Preface

With this book, we aim to bring the recent advances in nonlinear optics to the attention of the scientific community and present the reader with the bright spectrum of optics at the frontiers of research.

The eleven selected chapters in the book are authored by twenty-eight authors from ten different countries: Greece, Turkey, China, Romania, Brazil, Russia, India, Cameroon, Germany, and Italy.

Each chapter in this book is independent and self-contained, providing a contemporary overview of the methodology and approach used in theoretical and applied sciences. The reference list at the end of each chapter provides the reader with a selected list of journal papers, books, and conference proceedings.

The chapters can be summarized as follows:

In the first chapter, the two-dimensional nonlocal nonlinear Schrödinger model is studied and the connection between water waves and nonlinear optics is discussed.

In the second and third chapters, the derivative nonlinear Schrödinger equation (DNLS) with vanishing boundary conditions is studied using a novel approach in inverse scattering transform (IST). Based upon this new inverse scattering transform method, different kinds of soliton solutions such as the light/dark soliton, the pure soliton, and the mixed breather type are found for the derivative nonlinear Schrödinger equation with non-vanishing boundary conditions.

In the fourth chapter, the existence of magnetic solitons in atomic spinor Bose-Einstein condensates confined in an optical lattice are demonstrated.

The fifth chapter reviews the development of the Q-switched solid-state lasers, which are obtained using flash and diode pumping schemes based on active and passive Q-switchers.

In the sixth chapter, the laser cladding process is reported by proposing a semi-analytical model. The cladding process for different input powers and various laser beam velocities is studied.

In the seventh chapter, the main foundations for the conception, design, and the projection of optical sensors that explore the effects of nonlinear and quantum optics are presented.

In the eighth chapter, the digital sorting technique of vortex spectra is constructed. A new approach for measuring vortex spectra based on digital analyzing high-order intensity moments of complex beams is introduced.

In the ninth chapter, the main foundations of optical switches are reviewed and the design and fabrication of nonlinear optical switches are discussed from different aspects of the current technology.

In the tenth chapter, the evolution of light beams in a cubic-quintic-septic-nonical medium is investigated. As the model equation, an extended form of the well-known nonlinear Schrödinger equation is taken into account. By using a special ansatz, exact analytical solutions describing bright/dark and kink solitons are constructed.

In the last chapter, the authors investigated the modulational instability (MI) in Kerr media, induced by cross-phase modulation of two optical beams in nonlinear fiber including the effect of higher-order dispersion.

This book is intended to reach researchers, scientists, and postgraduate students in academia, as well as in industry, to give impressions and suggestions on a variety of topics and is published as an open access book in order to increase the impact of the contained information.

> **Dr. İlkay Bakırtaş and Dr. Nalan Antar** Department of Mathematics, Istanbul Technical University, Istanbul, Turkey

**Chapter 1**

**Abstract**

**1. Introduction**

Water Waves and Light: Two

*Dimitrios J. Frantzeskakis, Nalan Antar and İlkay Bakırtaş*

We study a generic model governing optical beam propagation in media featuring a nonlocal nonlinear response, namely a two-dimensional defocusing nonlocal nonlinear Schrödinger (NLS) model. Using a framework of multiscale expansions, the NLS model is reduced first to a bidirectional model, namely a Boussinesq or a Benney-Luke-type equation, and then to the unidirectional Kadomtsev-Petviashvili (KP) equation – both in Cartesian and cylindrical geometry. All the above models arise in the description of shallow water waves, and their solutions are used for the construction of relevant soliton solutions of the nonlocal NLS. Thus, the connection between water wave and nonlinear optics models suggests that patterns of water may indeed exist in light. We show that the NLS model supports intricate patterns that emerge from interactions between soliton stripes, as well as lump and ring

*Georgios N. Koutsokostas,Theodoros P. Horikis,*

solitons, similarly to the situation occurring in shallow water.

**Keywords:** Kadomtsev-Petviashvili equation, nonlocal nonlinear Schrödinger equation, line solitons, soliton interactions, lump solitons, patterns of light

small surface tension. In particular, the KP equation can be expressed as:

<sup>þ</sup> <sup>3</sup>*<sup>σ</sup> <sup>∂</sup>*<sup>2</sup> *u*

where *σ* ¼ �1: for *σ* ¼ þ1 (negative dispersion/small surface tension) Eq. (1) is a KPII, while for *σ* ¼ �1 (positive dispersion/large surface tension) Eq. (1) is a KPI.

*<sup>∂</sup>y*<sup>2</sup> <sup>¼</sup> 0, (1)

*∂ ∂x* *∂u ∂t* þ 6*u ∂u ∂x* þ *∂*3 *u ∂x*<sup>3</sup>

For many decades, solitons, namely robust localized waveforms that propagate undistorted in nonlinear dispersive media, have been a topic of particular interest in physics [1, 2] and applied mathematics [3, 4]. These waveforms have a variety of fascinating properties that should be mentioned: due to their particle-like nature, solitons collide elastically, thus preserving their shape after the collision process. In some cases, as e.g. in shallow water, line soliton collisions give rise to the emergence of various wave patterns, including X-, H-, or Y-shaped waves, as well as other, even more complicated, waveforms – see, e.g., Ref. [5] for a set of remarkable examples and observations. Such patterns can be described by multidimensional line soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation, a variant of the KP equation [6] with negative dispersion – as is the case of water waves with

Unlikely Partners

## **Chapter 1**

## Water Waves and Light: Two Unlikely Partners

*Georgios N. Koutsokostas,Theodoros P. Horikis, Dimitrios J. Frantzeskakis, Nalan Antar and İlkay Bakırtaş*

## **Abstract**

We study a generic model governing optical beam propagation in media featuring a nonlocal nonlinear response, namely a two-dimensional defocusing nonlocal nonlinear Schrödinger (NLS) model. Using a framework of multiscale expansions, the NLS model is reduced first to a bidirectional model, namely a Boussinesq or a Benney-Luke-type equation, and then to the unidirectional Kadomtsev-Petviashvili (KP) equation – both in Cartesian and cylindrical geometry. All the above models arise in the description of shallow water waves, and their solutions are used for the construction of relevant soliton solutions of the nonlocal NLS. Thus, the connection between water wave and nonlinear optics models suggests that patterns of water may indeed exist in light. We show that the NLS model supports intricate patterns that emerge from interactions between soliton stripes, as well as lump and ring solitons, similarly to the situation occurring in shallow water.

**Keywords:** Kadomtsev-Petviashvili equation, nonlocal nonlinear Schrödinger equation, line solitons, soliton interactions, lump solitons, patterns of light

### **1. Introduction**

For many decades, solitons, namely robust localized waveforms that propagate undistorted in nonlinear dispersive media, have been a topic of particular interest in physics [1, 2] and applied mathematics [3, 4]. These waveforms have a variety of fascinating properties that should be mentioned: due to their particle-like nature, solitons collide elastically, thus preserving their shape after the collision process. In some cases, as e.g. in shallow water, line soliton collisions give rise to the emergence of various wave patterns, including X-, H-, or Y-shaped waves, as well as other, even more complicated, waveforms – see, e.g., Ref. [5] for a set of remarkable examples and observations. Such patterns can be described by multidimensional line soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation, a variant of the KP equation [6] with negative dispersion – as is the case of water waves with small surface tension. In particular, the KP equation can be expressed as:

$$\frac{\partial}{\partial \mathbf{x}} \left( \frac{\partial u}{\partial t} + 6u \frac{\partial u}{\partial \mathbf{x}} + \frac{\partial^3 u}{\partial \mathbf{x}^3} \right) + 3\sigma \frac{\partial^2 u}{\partial y^2} = \mathbf{0},\tag{1}$$

where *σ* ¼ �1: for *σ* ¼ þ1 (negative dispersion/small surface tension) Eq. (1) is a KPII, while for *σ* ¼ �1 (positive dispersion/large surface tension) Eq. (1) is a KPI.

We should mention that both versions of the KP belong to the – rather limited – class of completely integrable equations in 2ð Þ þ 1 -dimensions [3].

Important to the collision-induced emergence of different types of wave patterns in shallow water is the fact that quasi (line) solitons of the KPII are stable. Furthermore, of paramount importance is the effect of soliton resonances [7–10], whereby two (or more) colliding solitons resonate under certain conditions, thus creating novel stable structures. It is, thus, not surprising that the collision dynamics of many robust solitons, can give rise to a wealth of complex wave patterns [11–16] (see also [1]).

**2. Nonlocal NLS and multiscale analysis**

*Water Waves and Light: Two Unlikely Partners DOI: http://dx.doi.org/10.5772/intechopen.95431*

**2.1 Introduction of the model and linear regime**

<sup>Δ</sup> � *<sup>∂</sup>*<sup>2</sup>

angle perturbation) [31, 32].

nonlocality.

**3**

The model under consideration is a two-dimensional (2D) defocusing NLS, with a nonlocal nonlinearity, which is expressed in the following dimensionless form:

Δ*u* � *nu* ¼ 0, (2)

1 *r*2 *∂*2

<sup>2</sup> <sup>¼</sup> <sup>0</sup>*:* (3)

*u* ¼ 0, (5)

*<sup>ρ</sup>* <sup>p</sup> exp ið Þ *<sup>ϕ</sup>* , (6)

*ρ<sup>t</sup>* þ **∇** � ð Þ¼ *ρ***∇***ϕ* 0, (8)

<sup>Δ</sup>*ρ*<sup>1</sup>*=*<sup>2</sup> <sup>¼</sup> 0, (7)

*<sup>θ</sup>*, (4)

2

i*ut* þ 1 2

where subscripts denote partial derivatives, *u*∈ , *n*∈ ,

*<sup>x</sup>* <sup>þ</sup> *<sup>∂</sup>*<sup>2</sup>

NLS equation with a local cubic (Kerr-type) nonlinearity:

*ϕ<sup>t</sup>* þ *n* þ

i*ut* þ 1 2

*d*Δ*n* � *n* þ j j *u*

*<sup>y</sup>* , or <sup>Δ</sup> � <sup>1</sup>

*r*

is the Laplacian in Cartesian or polar coordinates respectively, while the parameter *d*∈ <sup>þ</sup> measures the degree of nonlocality (see below). The above system finds a number of physical applications. For instance, in the context of optics, this model describes the evolution of the complex electric field envelope *u*, via a nonlinear wave equation in the paraxial regime, coupled with a diffusion-type equation for the nonlinear correction to the refractive index *n* depending on the intensity *I* ¼ j j *u*

(in this case, *t* represents the propagation coordinate) [25, 26]. In this context, the system (2)–(3) has been used to model experiments on liquid solutions exhibiting thermal nonlinearities [27, 28]. In addition, it has also been used in studies of plasmas (in this case, *n* represents the relative electron temperature perturbation) [29, 30], as well as in nematic liquid crystals (in this case, *n* is the optically induced

As mentioned above, parameter *d* represents the degree of nonlocality: for *d* ¼ 0, one recovers the local limit, whereby the system (2)–(3) reduces to the defocusing 2D

2

Δ*u* � j j *u*

that in our perturbative analytical approach below, *d* will be treated as a free parameter, of order *O*ð Þ1 , thus allowing the model to acquire an arbitrary degree of

model. This can be derived upon using the Madelung transformation:

*u* ¼ *u*<sup>0</sup>

1 2

while for *d* 6¼ 0, Eqs. (2) and (3) feature a spatially nonlocal nonlinearity. Notice

Our analytical approach relies on the analysis of the hydrodynamic form of the

ffiffiffi

where *u*<sup>0</sup> is a complex constant, while the unknown real functions *ρ* and *ϕ* denote the amplitude and phase of the field *u*, respectively. Substituting into Eqs. (2) and (3), we obtain the following hydrodynamical system of equations:

ð Þ **<sup>∇</sup>***<sup>ϕ</sup>* <sup>2</sup> � <sup>1</sup>

2 *ρ*�1*=*<sup>2</sup>

*<sup>∂</sup>r*ð Þþ *<sup>r</sup>∂<sup>r</sup>*

The above discussion motivates the following question: can solitons, similar to those governed by the KPII equation and feature rich collision dynamics (as in shallow water waves), be also predicted – and, possibly, observed – in nonlinear optics? In this context, key model is the nonlinear Schrödinger (NLS) equation, which is known to support soliton solutions (see, e.g., Ref. [17] and references therein). As was shown some time ago [18], the 2ð Þ þ 1 -dimensional defocusing NLS can be asymptotically reduced to the KPI equation, which allows one to approximate dark soliton stripes of the NLS as line solitons of KPI; then, the fact that line KPI solitons are unstable [3], can be used to investigate the transverse instability of dark solitons [18–20]. On the other hand, as was recently shown [21], the presence of nonlocality introduces an effective *surface tension* in the system. Then, in the case where the surface tension is small – corresponding to the case of *strong nonlocality* – it can be shown that the nonlocal NLS model can be asymptotically reduced to a *KPII equation*. This result allows for the derivation of approximate *stable* line soliton solutions of the nonlocal NLS, which behave similarly to the stable line solitons of KPII, i.e., they undergo elastic collisions and form patterns similar to those observed in shallow water.

The scope of this work is to review the above ideas and present a variety of soliton solutions and their dynamics in the framework of a defocusing nonlocal NLS model. These solutions stem from the soliton solutions of KP models, which are derived from the nonlocal NLS via a two-step multiscale expansion method. We show that, similarly to the water wave case, the nonlocal NLS can also support complicated wave patterns, arising from "complex interactions" between two or more antidark soliton stripes that are supported in the strong nonlocality regime. In addition, we present other soliton solutions, namely dark lumps (pertinent to the KPI equation), as well as ring dark and antidark solitons (pertinent to the cylindrical KP (cKP), *alias* Johnson's Equation [22, 23]). In particular, a summary of our main results, as well as the organization of this work, are as follows.

First, in Section 2, we introduce the nonlocal NLS model, present its continuouswave (cw) solution, and study its stability. Then, we perform a two-step multiscale analysis. First, at an intermediate stage, we derive a Boussinesq – or a Benney-Luke (BL) [24] – type equation, which is a long-wave, multi-dimensional and bidirectional, shallow water wave model. Then, from the Boussinesq/BL-type equation, we derive its far field, pertinent to the long time behavior, which is the KP equation, both in Cartesian and in cylindrical geometry. The reduction to the KP model allows us to construct approximate soliton solutions of the nonlocal NLS, which are presented in Section 3. In the strong nonlocality regime, we derive antidark soliton stripes, satisfying an effective KPII equation, which are shown to form patterns similar to those observed in shallow water; these include X-, Y, and H-shaped waveforms, as well as other complicated patterns arising from resonant interactions between more than two solitons. We also present dark lump solitons, as well as ring dark and antidark soliton solutions of the NLS, which are numerically found to propagate undistorted in the framework of the NLS model. Finally, in Section 4, we briefly discuss the conclusions of this work.

We should mention that both versions of the KP belong to the – rather limited – class of completely integrable equations in 2ð Þ þ 1 -dimensions [3].

[11–16] (see also [1]).

*Nonlinear Optics - From Solitons to Similaritons*

those observed in shallow water.

Important to the collision-induced emergence of different types of wave patterns in shallow water is the fact that quasi (line) solitons of the KPII are stable. Furthermore, of paramount importance is the effect of soliton resonances [7–10], whereby two (or more) colliding solitons resonate under certain conditions, thus creating novel stable structures. It is, thus, not surprising that the collision dynamics of many robust solitons, can give rise to a wealth of complex wave patterns

The above discussion motivates the following question: can solitons, similar to those governed by the KPII equation and feature rich collision dynamics (as in shallow water waves), be also predicted – and, possibly, observed – in nonlinear optics? In this context, key model is the nonlinear Schrödinger (NLS) equation, which is known to support soliton solutions (see, e.g., Ref. [17] and references therein). As was shown some time ago [18], the 2ð Þ þ 1 -dimensional defocusing NLS can be asymptotically reduced to the KPI equation, which allows one to approximate dark soliton stripes of the NLS as line solitons of KPI; then, the fact that line KPI solitons are unstable [3], can be used to investigate the transverse instability of dark solitons [18–20]. On the other hand, as was recently shown [21], the presence of nonlocality introduces an effective *surface tension* in the system. Then, in the case where the surface tension is small – corresponding to the case of *strong nonlocality* – it can be shown that the nonlocal NLS model can be asymptotically reduced to a *KPII equation*. This result allows for the derivation of approximate *stable* line soliton solutions of the nonlocal NLS, which behave similarly to the stable line solitons of KPII, i.e., they undergo elastic collisions and form patterns similar to

The scope of this work is to review the above ideas and present a variety of soliton solutions and their dynamics in the framework of a defocusing nonlocal NLS model. These solutions stem from the soliton solutions of KP models, which are derived from the nonlocal NLS via a two-step multiscale expansion method. We show that, similarly to the water wave case, the nonlocal NLS can also support complicated wave patterns, arising from "complex interactions" between two or more antidark soliton stripes that are supported in the strong nonlocality regime. In addition, we present other soliton solutions, namely dark lumps (pertinent to the KPI equation), as well as ring dark and antidark solitons (pertinent to the cylindrical KP (cKP), *alias* Johnson's Equation [22, 23]). In particular, a summary of our main

First, in Section 2, we introduce the nonlocal NLS model, present its continuouswave (cw) solution, and study its stability. Then, we perform a two-step multiscale analysis. First, at an intermediate stage, we derive a Boussinesq – or a Benney-Luke

(BL) [24] – type equation, which is a long-wave, multi-dimensional and bidirectional, shallow water wave model. Then, from the Boussinesq/BL-type equation, we derive its far field, pertinent to the long time behavior, which is the KP equation, both in Cartesian and in cylindrical geometry. The reduction to the KP model allows us to construct approximate soliton solutions of the nonlocal NLS, which are presented in Section 3. In the strong nonlocality regime, we derive antidark soliton stripes, satisfying an effective KPII equation, which are shown to form patterns similar to those observed in shallow water; these include X-, Y, and H-shaped waveforms, as well as other complicated patterns arising from resonant interactions between more than two solitons. We also present dark lump solitons, as well as ring dark and antidark soliton solutions of the NLS, which are numerically found to propagate undistorted in the framework of the NLS model. Finally, in

results, as well as the organization of this work, are as follows.

Section 4, we briefly discuss the conclusions of this work.

**2**

## **2. Nonlocal NLS and multiscale analysis**

#### **2.1 Introduction of the model and linear regime**

The model under consideration is a two-dimensional (2D) defocusing NLS, with a nonlocal nonlinearity, which is expressed in the following dimensionless form:

$$
\dot{\mathbf{u}}u\_t + \frac{1}{2}\Delta u - nu = \mathbf{0},\tag{2}
$$

$$\left|d\Delta n - n + \left|u\right|^2 = \mathbf{0}.\tag{3}$$

where subscripts denote partial derivatives, *u*∈ , *n*∈ ,

$$
\Delta \equiv \partial\_x^2 + \partial\_\chi^2, \quad \text{or} \quad \Delta \equiv \frac{1}{r} \partial\_r (r \partial\_r) + \frac{1}{r^2} \partial\_\theta^2, \tag{4}
$$

is the Laplacian in Cartesian or polar coordinates respectively, while the parameter *d*∈ <sup>þ</sup> measures the degree of nonlocality (see below). The above system finds a number of physical applications. For instance, in the context of optics, this model describes the evolution of the complex electric field envelope *u*, via a nonlinear wave equation in the paraxial regime, coupled with a diffusion-type equation for the nonlinear correction to the refractive index *n* depending on the intensity *I* ¼ j j *u* 2 (in this case, *t* represents the propagation coordinate) [25, 26]. In this context, the system (2)–(3) has been used to model experiments on liquid solutions exhibiting thermal nonlinearities [27, 28]. In addition, it has also been used in studies of plasmas (in this case, *n* represents the relative electron temperature perturbation) [29, 30], as well as in nematic liquid crystals (in this case, *n* is the optically induced angle perturbation) [31, 32].

As mentioned above, parameter *d* represents the degree of nonlocality: for *d* ¼ 0, one recovers the local limit, whereby the system (2)–(3) reduces to the defocusing 2D NLS equation with a local cubic (Kerr-type) nonlinearity:

$$
\dot{\mathbf{u}}u\_t + \frac{1}{2}\Delta u - |u|^2 u = \mathbf{0},\tag{5}
$$

while for *d* 6¼ 0, Eqs. (2) and (3) feature a spatially nonlocal nonlinearity. Notice that in our perturbative analytical approach below, *d* will be treated as a free parameter, of order *O*ð Þ1 , thus allowing the model to acquire an arbitrary degree of nonlocality.

Our analytical approach relies on the analysis of the hydrodynamic form of the model. This can be derived upon using the Madelung transformation:

$$
\mu = \mu\_0 \sqrt{\rho} \exp\left(\mathbf{i}\phi\right),
\tag{6}
$$

where *u*<sup>0</sup> is a complex constant, while the unknown real functions *ρ* and *ϕ* denote the amplitude and phase of the field *u*, respectively. Substituting into Eqs. (2) and (3), we obtain the following hydrodynamical system of equations:

$$
\phi\_t + n + \frac{1}{2} (\nabla \phi)^2 - \frac{1}{2} \rho^{-1/2} \Delta \rho^{1/2} = 0,\tag{7}
$$

$$
\rho\_t + \nabla \cdot (\rho \nabla \phi) = 0,\tag{8}
$$

$$d\Delta n - n + |u\_0|^2 \rho = 0,\tag{9}$$

where

Here, *c*<sup>2</sup>

identify the following two regimes:

solitons (that we will study in detail).

**2.2 The nonlinear regime – asymptotic analysis**

*2.2.1 The intermediate stage – Bousinesq/Benney-Luke equation*

• *α* <0 ) 4*d u*j j <sup>0</sup>

• *α* >0 ) 4*d u*j j <sup>0</sup>

**5**

*<sup>C</sup>*<sup>2</sup> <sup>¼</sup> j j *<sup>u</sup>*<sup>0</sup>

*α* ¼ 1 � 4*d u*j j <sup>0</sup>

on top of the cw solution. In addition, parameter *α* is given by:

of the dispersion relation for shallow water waves [4]:

*Water Waves and Light: Two Unlikely Partners DOI: http://dx.doi.org/10.5772/intechopen.95431*

> *ω*<sup>2</sup> ≈*k*<sup>2</sup> *c* 2 0 þ 1

2

2

2 0*h*2 *k*4

<sup>2</sup> <sup>&</sup>gt;1: weak surface tension – strong nonlocality (for fixed j j *<sup>u</sup>*<sup>0</sup>

<sup>2</sup> <sup>&</sup>lt;1: strong surface tension – weak nonlocality (for fixed j j *<sup>u</sup>*<sup>0</sup>

is the squared "speed of sound", namely the velocity of linear waves propagating

and plays the role of an effective surface tension for our original system. Indeed, as discussed in Ref. [21], this can be inferred by the fact that Eq. (17) is reminiscent

<sup>3</sup> <sup>3</sup>*T*^ � <sup>1</sup> *<sup>c</sup>*

water at rest), while *<sup>T</sup>*^ <sup>¼</sup> *<sup>T</sup><sup>=</sup> <sup>ρ</sup>gh*<sup>2</sup> , where *<sup>ρ</sup>* is the density and *<sup>T</sup>* the surface tension. Comparing Eqs. (17) and (20), one can identify the correspondence: 3*T*^ ! <sup>4</sup>*d u*j j <sup>0</sup>

Obviously, regarding the magnitude of the effective surface tension, one may

Below it will be shown that these two regimes correspond, respectively, to a KP-II and a KP-I equation, with the former giving rise to complex interactions of line

We now proceed by analyzing the fully nonlinear regime. We start by seeking

2 *<sup>t</sup>* <sup>þ</sup> *<sup>ε</sup>*<sup>1</sup>*=*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> *<sup>ε</sup>n*<sup>1</sup> <sup>þ</sup> *<sup>ε</sup>*<sup>2</sup>

where 0 <*ε* ≪ 1 is a formal small parameter and the unknown functions *ρ <sup>j</sup>*

*<sup>x</sup>*, *<sup>Y</sup>* <sup>¼</sup> *<sup>ε</sup>*<sup>1</sup>*=*<sup>2</sup>

*<sup>y</sup>*, *<sup>T</sup>* <sup>¼</sup> *<sup>ε</sup>*<sup>1</sup>*=*<sup>2</sup>

*<sup>R</sup>* <sup>¼</sup> *<sup>ε</sup>*<sup>1</sup>*=*<sup>2</sup>*r*, *<sup>θ</sup>* <sup>¼</sup> *<sup>θ</sup>*, *<sup>T</sup>* <sup>¼</sup> *<sup>ε</sup>*<sup>1</sup>*=*<sup>2</sup>*t*, (25)

solutions of Eqs. (7)–(9) in the form of the following asymptotic expansions:

*<sup>ρ</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> *ερ*<sup>1</sup> <sup>þ</sup> *<sup>ε</sup>*<sup>2</sup>

*ϕ* ¼ �j j *u*<sup>0</sup>

*n* ¼ j j *u*<sup>0</sup>

and *n <sup>j</sup>* (*j* ¼ 1, 2, … ) depend on the following stretched variables:

*<sup>X</sup>* <sup>¼</sup> *<sup>ε</sup>*<sup>1</sup>*=*<sup>2</sup>

which indeed suggests that a *surface tension* analogue in our problem is ∝*d u*j j <sup>0</sup>

depending on the nonlocality parameter *d* and the cw intensity j j *u*<sup>0</sup>

<sup>0</sup> ¼ *gh* is the velocity (*g* is the acceleration of gravity and *h* the depth of

, (18)

, (19)

*:* (20)

2 .

*ρ*<sup>2</sup> þ ⋯, (21)

Φ (22)

*t*, (24)

*n*<sup>2</sup> þ ⋯, (23)

2 ,

2 ),

2 ).

, Φ

2 ,

where **<sup>∇</sup>** � *<sup>∂</sup>x*, *<sup>∂</sup><sup>y</sup>* or **<sup>∇</sup>** � *<sup>∂</sup>r*, <sup>1</sup> *r ∂θ* is the gradient operator in Cartesian or in polar coordinates respectively.

The simplest, non trivial, solution of Eqs. (2) and (3) is of the form:

$$\rho = \mathbf{1}, \quad \phi = -|u\_0|^2 t, \quad n = |u\_0|^2,\tag{10}$$

which corresponds to the continuous wave (cw) solution *u* ¼ *u*<sup>0</sup> exp ð�ij j *u*<sup>0</sup> 2 *t*Þ and the constant function *n* ¼ j j *u*<sup>0</sup> 2 . This solution will serve as a "pedestal", on top of which we will seek nonlinear excitations, namely solitons. It is thus necessary to investigate whether the solution (7) is modulationally stable. This can be done upon introducing to Eqs. (7)–(9) the following perturbation ansatz:

$$
\rho = \mathbf{1} + \epsilon \tilde{\rho}, \quad \phi = -|u\_0|^2 t + \epsilon \tilde{\phi}, \quad n = \left| u\_0 \right|^2 + \epsilon \tilde{n}, \tag{11}
$$

where 0 <*ε* ≪ 1 is a formal small parameter. This way, we obtain, at *O*ð Þ*ε* , the following linear system:

$$
\ddot{\phi}\_t + \ddot{n} - \frac{1}{4}\Delta\ddot{\rho} = 0,\tag{12}
$$

$$
\tilde{\rho}\_t + \Delta \tilde{\phi} = \mathbf{0},
\tag{13}
$$

$$d\Delta\ddot{n} - \ddot{n} + |u\_0|^2 \ddot{\rho} = 0.\tag{14}$$

The above system can straightforwardly be decoupled as follows: solving Eq. (12) for *n*~, as well as Eq. (13) for Δ*ϕ*~, and substituting back in Eq. (13), the following linear equation for ~*ρ* is obtained:

$$\left|\tilde{\rho}\_{\rm tt} - \left|u\_0\right|^2 \Delta \tilde{\rho} - d\Delta \tilde{\rho}\_{\rm tt} + \frac{1}{4} (\mathbf{1} - d\Delta) \Delta^2 \tilde{\rho} = \mathbf{0}.\tag{15}$$

Since the above solution is linear we can examine its stability in first order by examining the dispersion relation of plane wave solutions, i.e., ∝exp i½ � ð Þ *k* � *r* � *ωt* , of wavenumber *k* ¼ k*k*k and frequency *ω* (here, e.g., in Cartesian geometry, *r* ¼ ð Þ *x*, *y* and *k* ¼ *kx*, *ky* ). Then, it can readily be found that these plane waves are characterized by the following dispersion relation:

$$
\alpha^2 = \frac{|u\_0|^2 k^2}{1 + dk^2} + \frac{1}{4}k^4. \tag{16}
$$

From the above equation we can obtain the following information. First, it is observed that *ω*∈ ∀*k*∈ , which indicates that the steady-state solution is modulationally stable. This result is important since, below, we will seek for soliton solutions on top of the stable cw background (7). Second, in the long-wavelength limit, such that *dk*<sup>2</sup> ≪ 1, Eq. (16) can be reduced to the following Bogoliubov-type dispersion relation:

$$
\omega \alpha^2 \approx k^2 C^2 + \frac{1}{4} a k^4,\tag{17}
$$

*Water Waves and Light: Two Unlikely Partners DOI: http://dx.doi.org/10.5772/intechopen.95431*

where

*d*Δ*n* � *n* þ j j *u*<sup>0</sup>

*r ∂θ*

*ρ* ¼ 1, *ϕ* ¼ �j j *u*<sup>0</sup>

2

introducing to Eqs. (7)–(9) the following perturbation ansatz:

*ρ* ¼ 1 þ *ε*~*ρ*, *ϕ* ¼ �j j *u*<sup>0</sup>

The simplest, non trivial, solution of Eqs. (2) and (3) is of the form:

where **<sup>∇</sup>** � *<sup>∂</sup>x*, *<sup>∂</sup><sup>y</sup>*

following linear system:

*r* ¼ ð Þ *x*, *y* and *k* ¼ *kx*, *ky*

dispersion relation:

**4**

polar coordinates respectively.

and the constant function *n* ¼ j j *u*<sup>0</sup>

following linear equation for ~*ρ* is obtained:

~*ρtt* � j j *u*<sup>0</sup>

characterized by the following dispersion relation:

2

or **<sup>∇</sup>** � *<sup>∂</sup>r*, <sup>1</sup>

*Nonlinear Optics - From Solitons to Similaritons*

2

2

which corresponds to the continuous wave (cw) solution *u* ¼ *u*<sup>0</sup> exp ð�ij j *u*<sup>0</sup>

of which we will seek nonlinear excitations, namely solitons. It is thus necessary to investigate whether the solution (7) is modulationally stable. This can be done upon

2

where 0 <*ε* ≪ 1 is a formal small parameter. This way, we obtain, at *O*ð Þ*ε* , the

4

2

1

Since the above solution is linear we can examine its stability in first order by examining the dispersion relation of plane wave solutions, i.e., ∝exp i½ � ð Þ *k* � *r* � *ωt* , of wavenumber *k* ¼ k*k*k and frequency *ω* (here, e.g., in Cartesian geometry,

> 2 *k*2 <sup>1</sup> <sup>þ</sup> *dk*<sup>2</sup> <sup>þ</sup>

From the above equation we can obtain the following information. First, it is

modulationally stable. This result is important since, below, we will seek for soliton solutions on top of the stable cw background (7). Second, in the long-wavelength limit, such that *dk*<sup>2</sup> ≪ 1, Eq. (16) can be reduced to the following Bogoliubov-type

> *<sup>C</sup>*<sup>2</sup> <sup>þ</sup> 1 <sup>4</sup> *<sup>α</sup>k*<sup>4</sup>

<sup>4</sup> ð Þ <sup>1</sup> � *<sup>d</sup>*<sup>Δ</sup> <sup>Δ</sup><sup>2</sup>

). Then, it can readily be found that these plane waves are

1 4 *k*4

*<sup>ϕ</sup>*~*<sup>t</sup>* <sup>þ</sup> *<sup>n</sup>*<sup>~</sup> � <sup>1</sup>

*d*Δ*n*~ � *n*~ þ j j *u*<sup>0</sup>

Δ~*ρ* � *d*Δ~*ρtt* þ

*<sup>ω</sup>*<sup>2</sup> <sup>¼</sup> j j *<sup>u</sup>*<sup>0</sup>

observed that *ω*∈ ∀*k*∈ , which indicates that the steady-state solution is

*ω*<sup>2</sup> ≈ *k*<sup>2</sup>

The above system can straightforwardly be decoupled as follows: solving Eq. (12) for *n*~, as well as Eq. (13) for Δ*ϕ*~, and substituting back in Eq. (13), the

*t*, *n* ¼ j j *u*<sup>0</sup>

*<sup>t</sup>* <sup>þ</sup> *εϕ*~, *<sup>n</sup>* <sup>¼</sup> j j *<sup>u</sup>*<sup>0</sup>

*ρ* ¼ 0, (9)

, (10)

<sup>2</sup> <sup>þ</sup> *<sup>ε</sup>n*~, (11)

2 *t*Þ

is the gradient operator in Cartesian or in

2

. This solution will serve as a "pedestal", on top

Δ~*ρ* ¼ 0, (12)

~*ρ* ¼ 0*:* (14)

~*ρ* ¼ 0*:* (15)

*:* (16)

, (17)

<sup>~</sup>*ρ<sup>t</sup>* <sup>þ</sup> <sup>Δ</sup>*ϕ*<sup>~</sup> <sup>¼</sup> 0, (13)

$$\mathbf{C}^2 = |\boldsymbol{u}\_0|^2,\tag{18}$$

is the squared "speed of sound", namely the velocity of linear waves propagating on top of the cw solution. In addition, parameter *α* is given by:

$$a = 1 - 4d|u\_0|^2,\tag{19}$$

and plays the role of an effective surface tension for our original system. Indeed, as discussed in Ref. [21], this can be inferred by the fact that Eq. (17) is reminiscent of the dispersion relation for shallow water waves [4]:

$$
\hbar \alpha^2 \approx k^2 c\_0^2 + \frac{1}{3} \left( 3\hat{T} - \mathbf{1} \right) c\_0^2 h^2 k^4. \tag{20}
$$

Here, *c*<sup>2</sup> <sup>0</sup> ¼ *gh* is the velocity (*g* is the acceleration of gravity and *h* the depth of water at rest), while *<sup>T</sup>*^ <sup>¼</sup> *<sup>T</sup><sup>=</sup> <sup>ρ</sup>gh*<sup>2</sup> , where *<sup>ρ</sup>* is the density and *<sup>T</sup>* the surface tension. Comparing Eqs. (17) and (20), one can identify the correspondence: 3*T*^ ! <sup>4</sup>*d u*j j <sup>0</sup> 2 , which indeed suggests that a *surface tension* analogue in our problem is ∝*d u*j j <sup>0</sup> 2 , depending on the nonlocality parameter *d* and the cw intensity j j *u*<sup>0</sup> 2 .

Obviously, regarding the magnitude of the effective surface tension, one may identify the following two regimes:


Below it will be shown that these two regimes correspond, respectively, to a KP-II and a KP-I equation, with the former giving rise to complex interactions of line solitons (that we will study in detail).

#### **2.2 The nonlinear regime – asymptotic analysis**

#### *2.2.1 The intermediate stage – Bousinesq/Benney-Luke equation*

We now proceed by analyzing the fully nonlinear regime. We start by seeking solutions of Eqs. (7)–(9) in the form of the following asymptotic expansions:

$$
\rho = \mathbf{1} + \varepsilon \rho\_1 + \varepsilon^2 \rho\_2 + \cdots,\tag{21}
$$

$$\phi = -|u\_0|^2 t + \varepsilon^{1/2} \Phi \tag{22}$$

$$
\hbar n = |\mu\_0|^2 + \epsilon n\_1 + \epsilon^2 n\_2 + \cdots,\tag{23}
$$

where 0 <*ε* ≪ 1 is a formal small parameter and the unknown functions *ρ <sup>j</sup>* , Φ and *n <sup>j</sup>* (*j* ¼ 1, 2, … ) depend on the following stretched variables:

$$X = e^{1/2} \mathfrak{x}, \quad Y = e^{1/2} \mathfrak{y}, \quad T = e^{1/2} \mathfrak{t}, \tag{24}$$

$$R = \varepsilon^{1/2}r, \quad \theta = \theta, \quad T = \varepsilon^{1/2}t,\tag{25}$$

for the Cartesian and polar geometry, respectively (the angular coordinate *θ* in the polar geometry remains unchanged). By substituting the expansions (21)–(23) into Eqs. (7)–(9), and equating terms of the same order in *ε*, we obtain the following results. First, Eq. (13), at O *ε*3*=*<sup>2</sup> and O *ε*5*=*<sup>2</sup> , yields:

$$
\rho\_{1T} + \tilde{\Delta}\Phi = \mathbf{0}, \quad \rho\_{2T} + \tilde{\mathbf{V}} \cdot \left(\rho\_1 \tilde{\mathbf{V}} \Phi\right) = \mathbf{0}. \tag{26}
$$

Φ0*χ*~*<sup>χ</sup>* ¼ 0, (34) Φ0*ϱϱ*<sup>~</sup> ¼ 0, (35)

<sup>0</sup> , which depends on *χ*, and a left-going one,

<sup>0</sup> ð Þ <sup>~</sup>*χ*,Y, <sup>T</sup> , (36)

<sup>0</sup> ð Þ *<sup>ϱ</sup>*~, <sup>Θ</sup>, <sup>T</sup> *:* (37)

(38)

(39)

<sup>0</sup> and <sup>Φ</sup>ð Þ *<sup>L</sup>*

<sup>1</sup>YY ¼ 0, (40)

<sup>1</sup>YY ¼ 0, (41)

<sup>0</sup> in

� *<sup>C</sup>*<sup>2</sup> Φð Þ *<sup>R</sup>*

� *<sup>C</sup>*<sup>2</sup> Φð Þ *<sup>L</sup>*

<sup>T</sup> <sup>Φ</sup>ð Þ *<sup>R</sup>* 0*ϱ*~

<sup>T</sup> <sup>Φ</sup>ð Þ *<sup>L</sup>* 0*ϱ*~

<sup>1</sup> satisfying a pair of two uncoupled KP equations.

þ *C* 2 *ρ* ð Þ *R*

� *C* 2 *ρ* ð Þ *L* � 1 <sup>T</sup> <sup>2</sup> <sup>Φ</sup>ð Þ *<sup>R</sup>*

� 1 <sup>T</sup> <sup>2</sup> <sup>Φ</sup>ð Þ *<sup>R</sup>*

in Cartesian and polar coordinates respectively. The above equations imply that Φ<sup>0</sup> can be expressed as a superposition of two waves. In the Cartesian geometry,

<sup>0</sup> , depending on ~*χ*. Similarly, in the polar case, Φ<sup>0</sup> can be expressed as superposition of a radially expanding wave (depending on *ϱ*), and a radially contracting one

<sup>0</sup> ð Þþ *<sup>χ</sup>*, <sup>Y</sup>, <sup>T</sup> <sup>Φ</sup>ð Þ *<sup>L</sup>*

<sup>0</sup> ð Þþ *<sup>ϱ</sup>*, <sup>Θ</sup>, <sup>T</sup> <sup>Φ</sup>ð Þ *<sup>L</sup>*

<sup>0</sup>*<sup>χ</sup>* <sup>Φ</sup>ð Þ *<sup>L</sup>* 0~*χ*~*χ*

3*C* 2 Φð Þ *<sup>L</sup>* <sup>2</sup> <sup>0</sup>*ϱ*<sup>~</sup> � *<sup>C</sup>*

for the polar geometry. Here, it is important to observe the following. Once Eq. (38) is integrated in *χ* or ~*χ* [and, similarly, Eq. (39) in *ϱ* or *ϱ*~], secular terms arise: the secular therms are those in the square brackets in the right-hand side, because are functions of *χ* or ~*χ* (*ϱ* or ~*ϱ* in polar) alone, not both. Removal of these secular

both geometries. Furthermore, using the equation Φ*<sup>T</sup>* ¼ �*n*<sup>1</sup> (i.e., the linear part of Eq. (29), together with Eqs. (26) and (27), it is found that the amplitude *ρ*<sup>1</sup> can also

<sup>0</sup>YY

<sup>0</sup>YY ,

<sup>0</sup>ΘΘ

<sup>0</sup>ΘΘ ,

3*C* 2 Φð Þ *<sup>L</sup>* <sup>2</sup> 0~~*χ*

(depending on *ϱ*~). Thus, the solutions of Eqs. (34) and (35) read:

In addition, at order Oð Þ*ε* , we obtain the following equations:

<sup>0</sup>~*<sup>χ</sup>* � <sup>Φ</sup>ð Þ *<sup>R</sup>*

<sup>0</sup><sup>T</sup> <sup>þ</sup> *<sup>α</sup>* 4 Φð Þ *<sup>R</sup>* <sup>0</sup>*χχχ* � <sup>3</sup>*<sup>C</sup>* 2 Φð Þ *<sup>R</sup>* <sup>2</sup> 0*χ*

<sup>0</sup>*<sup>ϱ</sup>* <sup>Φ</sup>ð Þ *<sup>L</sup>* 0~*ϱϱ*~

terms leads to two uncoupled nonlinear evolution equations for Φð Þ *<sup>R</sup>*

be decomposed to a left- and a right-going wave; this means that

ð Þ *L*

In the Cartesian geometry, the KP equations are of the form:

*ρ*<sup>1</sup> ¼ *ρ* ð Þ *R* <sup>1</sup> þ *ρ* ð Þ *L* <sup>1</sup> ,

3*C* 2 *ρ* ð Þ *R* <sup>1</sup> *ρ* ð Þ *R* 1*χ*

<sup>0</sup><sup>T</sup> <sup>þ</sup> *<sup>α</sup>* 4 Φð Þ *<sup>L</sup>* <sup>0</sup>~*χ*~*χ*~*<sup>χ</sup>* þ

<sup>Φ</sup><sup>0</sup> <sup>¼</sup> <sup>Φ</sup>ð Þ *<sup>R</sup>*

<sup>Φ</sup><sup>0</sup> <sup>¼</sup> <sup>Φ</sup>ð Þ *<sup>R</sup>*

0*χχ*Φð Þ *<sup>L</sup>*

<sup>þ</sup> *<sup>∂</sup><sup>χ</sup>* �2*C*Φð Þ *<sup>R</sup>*

<sup>þ</sup> *<sup>∂</sup>*~*<sup>χ</sup>* <sup>2</sup>*C*Φð Þ *<sup>L</sup>*

0ϱϱΦð Þ *<sup>L</sup>*

<sup>þ</sup> *<sup>∂</sup><sup>ρ</sup>* �2*C*Φð Þ *<sup>R</sup>*

<sup>þ</sup> *<sup>∂</sup>ϱ*<sup>~</sup> <sup>2</sup>*C*Φð Þ *<sup>L</sup>*

ð Þ *R* <sup>1</sup> and *ρ*

*∂<sup>χ</sup> ρ* ð Þ *R* <sup>1</sup><sup>T</sup> � *<sup>α</sup>* <sup>8</sup>*C<sup>ρ</sup>* ð Þ *R* <sup>1</sup>*χχχ* þ

*∂*~*<sup>χ</sup> ρ* ð Þ *L* <sup>1</sup><sup>T</sup> <sup>þ</sup> *<sup>α</sup>* <sup>8</sup>*C<sup>ρ</sup>* ð Þ *L* <sup>1</sup>~*χ*~*χ*~*<sup>χ</sup>* � <sup>3</sup>*<sup>C</sup>* 2 *ρ* ð Þ *L* <sup>1</sup> *ρ* ð Þ *L* 1~*χ*

<sup>0</sup>~*<sup>ϱ</sup>* � <sup>Φ</sup>ð Þ *<sup>R</sup>*

<sup>0</sup><sup>T</sup> <sup>þ</sup> *<sup>α</sup>* 4 Φð Þ *<sup>R</sup>* <sup>0</sup>ϱϱϱ � <sup>3</sup>*<sup>C</sup>* 2 Φð Þ *<sup>R</sup>* <sup>2</sup> <sup>0</sup>*<sup>ϱ</sup>* � *<sup>C</sup>*

<sup>0</sup><sup>T</sup> <sup>þ</sup> *<sup>α</sup>* 4 Φð Þ *<sup>L</sup>* <sup>0</sup>*ϱ*~*ϱ*~*ϱ*<sup>~</sup> þ

these waves are a right-going one, Φð Þ *<sup>R</sup>*

*Water Waves and Light: Two Unlikely Partners DOI: http://dx.doi.org/10.5772/intechopen.95431*

<sup>Φ</sup>1*χ*~*<sup>χ</sup>* <sup>¼</sup> �*<sup>C</sup>* <sup>Φ</sup>ð Þ *<sup>R</sup>*

for the Cartesian geometry, and

<sup>Φ</sup>1*ϱϱ*<sup>~</sup> <sup>¼</sup> �*<sup>C</sup>* <sup>Φ</sup>ð Þ *<sup>R</sup>*

with the fields *ρ*

**7**

Φð Þ *<sup>L</sup>*

4*C*<sup>2</sup>

4*C*<sup>2</sup>

Furthermore, Eq. (14) at <sup>O</sup>ð Þ*<sup>ε</sup>* and <sup>O</sup> *<sup>ε</sup>*<sup>2</sup> ð Þ leads to the equations:

$$
\mu\_1 = \left| u\_0 \right|^2 \rho\_1, \quad d\bar{\Delta}n\_1 - n\_2 + \left| u\_0 \right|^2 \rho\_2 = \mathbf{0}, \tag{27}
$$

which connect the amplitudes *ρ*1,2 and *n*1,2 with the phase Φ; note that, here,

$$\tilde{\Delta} \equiv \partial\_X^2 + \partial\_Y^2, \quad \tilde{\nabla} \equiv (\partial\_X, \partial\_Y), \text{ or } \tilde{\Delta} \equiv \frac{1}{R} \partial\_R (R \partial\_R) + \frac{1}{R^2} \partial\_\theta^2, \quad \tilde{\nabla} \equiv \left(\partial\_R, \frac{1}{R} \partial\_\theta\right), \tag{28}$$

in Cartesian and polar coordinates, respectively. In addition, Eq. (14) yields:

$$
\boldsymbol{\Phi}\_{T} + \boldsymbol{n}\_{1} + \varepsilon \left[ \frac{1}{2} \partial\_{T} \left( \tilde{\boldsymbol{\nabla}} \boldsymbol{\Phi} \right)^{2} + \boldsymbol{n}\_{2} - \frac{1}{4} \tilde{\boldsymbol{\Delta}} \boldsymbol{\rho}\_{1} \right] = \boldsymbol{\mathcal{O}} \left( \varepsilon^{2} \right). \tag{29}$$

To this end, combining Eq. (29) with Eqs. (26) and (27), yields the following biderectional nonlinear dispersive wave equation for Φ:

$$
\boldsymbol{\Phi}\_{TT} - \boldsymbol{\mathcal{C}}^2 \boldsymbol{\tilde{\Delta}} \boldsymbol{\Phi} + \varepsilon \left[ \frac{1}{4} a \boldsymbol{\tilde{\Delta}}^2 \boldsymbol{\Phi} + \frac{1}{2} \boldsymbol{\partial\_T} (\boldsymbol{\tilde{\nabla}} \boldsymbol{\Phi})^2 + \boldsymbol{\tilde{\nabla}} \cdot \left( \boldsymbol{\Phi}\_T \boldsymbol{\tilde{\nabla}} \boldsymbol{\Phi} \right) \right] = \boldsymbol{\mathcal{O}} (\varepsilon^2), \tag{30}
$$

where *C*<sup>2</sup> is the squared speed of sound given by Eq. (18) and the parameter *α* is given by Eq. (19). Here it is worth observing the following. First, at the leading-order in *ε*, Eq. (30) is the standard second-order wave equation, while at order Oð Þ*ε* , the linear part of Eq. (30) corresponds to the dispersion relation (16) for the smallamplitude linear waves of Eqs. (2) and (3) propagating on top of the steady state with ∣*u*∣ ¼ ∣*u*0∣ and *n* ¼ j j *u*<sup>0</sup> 2 . The full Eq. (30), incorporates fourth-order dispersion and quadratic nonlinear terms, resembling the Boussinesq and Benney-Luke [24] equations. These models, are used to describe bidirectional shallow water waves, in the framework of small-amplitude and long-wave approximations [4].

#### *2.2.2 Long-time behavior – KP equation*

Using a multiscale expansion method, similar to the one employed in the water wave problem [4], we now derive the KP equation, which is obtained under the additional assumptions of *quasi-two-dimensionality* and *unidirectional propagation*. In particular, we introduce the asymptotic expansion:

$$
\Phi = \Phi\_0 + \varepsilon \,\Phi\_1 + \cdots,\tag{31}
$$

where the unknown functions Φ<sup>ℓ</sup> (ℓ ¼ 0, 1, … ) depend on the variables:

$$\chi = \mathbf{X} - \mathbf{C}\mathbf{T}, \quad \tilde{\chi} = \mathbf{X} + \mathbf{C}\mathbf{T}, \quad \mathcal{Y} = \varepsilon^{1/2}\mathbf{Y}, \quad \mathcal{T} = \varepsilon\mathbf{T}, \tag{32}$$

$$
\varrho = \mathbb{R} - \mathbb{C}T, \quad \tilde{\varrho} = \mathbb{R} + \mathbb{C}T, \quad \Theta = \varepsilon^{-1/2}\theta, \quad T = \varepsilon T,\tag{33}
$$

for the Cartesian and polar geometry, respectively. Substituting the expansion (31) into Eq. (30), at the leading-order in *ε*, we obtain the equations:

*Water Waves and Light: Two Unlikely Partners DOI: http://dx.doi.org/10.5772/intechopen.95431*

for the Cartesian and polar geometry, respectively (the angular coordinate *θ* in the polar geometry remains unchanged). By substituting the expansions (21)–(23) into Eqs. (7)–(9), and equating terms of the same order in *ε*, we obtain the follow-

*<sup>ρ</sup>*1, *<sup>d</sup>*Δ~*n*<sup>1</sup> � *<sup>n</sup>*<sup>2</sup> <sup>þ</sup> j j *<sup>u</sup>*<sup>0</sup>

which connect the amplitudes *ρ*1,2 and *n*1,2 with the phase Φ; note that, here,

in Cartesian and polar coordinates, respectively. In addition, Eq. (14) yields:

To this end, combining Eq. (29) with Eqs. (26) and (27), yields the following

*<sup>∂</sup><sup>T</sup>* **<sup>∇</sup>**~<sup>Φ</sup> <sup>2</sup> <sup>þ</sup> **<sup>∇</sup>**<sup>~</sup> � <sup>Φ</sup>*T***∇**~<sup>Φ</sup>

quadratic nonlinear terms, resembling the Boussinesq and Benney-Luke [24] equations. These models, are used to describe bidirectional shallow water waves, in the

Using a multiscale expansion method, similar to the one employed in the water wave problem [4], we now derive the KP equation, which is obtained under the additional assumptions of *quasi-two-dimensionality* and *unidirectional propagation*.

where the unknown functions Φ<sup>ℓ</sup> (ℓ ¼ 0, 1, … ) depend on the variables:

for the Cartesian and polar geometry, respectively. Substituting the expansion

*<sup>ϱ</sup>* <sup>¼</sup> *<sup>R</sup>* � *CT*, *<sup>ϱ</sup>*<sup>~</sup> <sup>¼</sup> *<sup>R</sup>* <sup>þ</sup> *CT*, <sup>Θ</sup> <sup>¼</sup> *<sup>ε</sup>*�1*=*<sup>2</sup>*θ*, <sup>T</sup> <sup>¼</sup> *<sup>ε</sup>T*, (33)

*<sup>χ</sup>* <sup>¼</sup> *<sup>X</sup>* � *CT*, <sup>~</sup>*<sup>χ</sup>* <sup>¼</sup> *<sup>X</sup>* <sup>þ</sup> *CT*, <sup>Y</sup> <sup>¼</sup> *<sup>ε</sup>*<sup>1</sup>*=*<sup>2</sup>

(31) into Eq. (30), at the leading-order in *ε*, we obtain the equations:

where *C*<sup>2</sup> is the squared speed of sound given by Eq. (18) and the parameter *α* is given by Eq. (19). Here it is worth observing the following. First, at the leading-order in *ε*, Eq. (30) is the standard second-order wave equation, while at order Oð Þ*ε* , the linear part of Eq. (30) corresponds to the dispersion relation (16) for the smallamplitude linear waves of Eqs. (2) and (3) propagating on top of the steady state with

*<sup>R</sup> <sup>∂</sup>R*ð Þþ *<sup>R</sup>∂<sup>R</sup>*

<sup>þ</sup> *<sup>n</sup>*<sup>2</sup> � <sup>1</sup> 4 Δ~*ρ*<sup>1</sup>

*<sup>ρ</sup>*1*<sup>T</sup>* <sup>þ</sup> ΔΦ<sup>~</sup> <sup>¼</sup> 0, *<sup>ρ</sup>*2*<sup>T</sup>* <sup>þ</sup> **<sup>∇</sup>**<sup>~</sup> � *<sup>ρ</sup>*1**∇**~<sup>Φ</sup> <sup>¼</sup> <sup>0</sup>*:* (26)

2

1 *R*2 *∂*2

. The full Eq. (30), incorporates fourth-order dispersion and

Φ ¼ Φ<sup>0</sup> þ *ε* Φ<sup>1</sup> þ ⋯, (31)

*Y*, T ¼ *εT*, (32)

*ρ*<sup>2</sup> ¼ 0, (27)

1 *R ∂θ* 

<sup>¼</sup> <sup>O</sup> *<sup>ε</sup>*<sup>2</sup> *:* (29)

<sup>¼</sup> <sup>O</sup> *<sup>ε</sup>*<sup>2</sup> , (30)

, (28)

*<sup>θ</sup>*, **<sup>∇</sup>**<sup>~</sup> � *<sup>∂</sup>R*,

ing results. First, Eq. (13), at O *ε*3*=*<sup>2</sup> and O *ε*5*=*<sup>2</sup> , yields:

*Nonlinear Optics - From Solitons to Similaritons*

*n*<sup>1</sup> ¼ j j *u*<sup>0</sup>

*<sup>Y</sup>*, **<sup>∇</sup>**<sup>~</sup> � ð Þ *<sup>∂</sup>X*, *<sup>∂</sup><sup>Y</sup>* , or <sup>Δ</sup><sup>~</sup> � <sup>1</sup>

<sup>Φ</sup>*<sup>T</sup>* <sup>þ</sup> *<sup>n</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ε</sup>* <sup>1</sup>

ΔΦ<sup>~</sup> <sup>þ</sup> *<sup>ε</sup>* <sup>1</sup>

2

*2.2.2 Long-time behavior – KP equation*

biderectional nonlinear dispersive wave equation for Φ:

<sup>4</sup> *<sup>α</sup>*Δ~<sup>2</sup>

<sup>Δ</sup><sup>~</sup> � *<sup>∂</sup>*<sup>2</sup>

*<sup>X</sup>* <sup>þ</sup> *<sup>∂</sup>*<sup>2</sup>

<sup>Φ</sup>*TT* � *<sup>C</sup>*<sup>2</sup>

∣*u*∣ ¼ ∣*u*0∣ and *n* ¼ j j *u*<sup>0</sup>

**6**

Furthermore, Eq. (14) at <sup>O</sup>ð Þ*<sup>ε</sup>* and <sup>O</sup> *<sup>ε</sup>*<sup>2</sup> ð Þ leads to the equations:

2

2

Φ þ 1 2

framework of small-amplitude and long-wave approximations [4].

In particular, we introduce the asymptotic expansion:

*<sup>∂</sup><sup>T</sup>* **<sup>∇</sup>**~<sup>Φ</sup> <sup>2</sup>

$$
\Phi\_{0\chi\hat{\chi}} = 0,\tag{34}
$$

$$
\Phi\_{0q\bar{q}} = 0,\tag{35}
$$

in Cartesian and polar coordinates respectively. The above equations imply that Φ<sup>0</sup> can be expressed as a superposition of two waves. In the Cartesian geometry, these waves are a right-going one, Φð Þ *<sup>R</sup>* <sup>0</sup> , which depends on *χ*, and a left-going one, Φð Þ *<sup>L</sup>* <sup>0</sup> , depending on ~*χ*. Similarly, in the polar case, Φ<sup>0</sup> can be expressed as superposition of a radially expanding wave (depending on *ϱ*), and a radially contracting one (depending on *ϱ*~). Thus, the solutions of Eqs. (34) and (35) read:

$$\Phi\_0 = \Phi\_0^{(\mathcal{R})}(\boldsymbol{\chi}, \mathcal{Y}, \mathcal{T}) + \Phi\_0^{(L)}(\boldsymbol{\tilde{\chi}}, \mathcal{Y}, \mathcal{T}), \tag{36}$$

$$
\Phi\_0 = \Phi\_0^{(\mathcal{R})}(\varrho, \Theta, \mathcal{T}) + \Phi\_0^{(L)}(\tilde{\varrho}, \Theta, \mathcal{T}).\tag{37}
$$

In addition, at order Oð Þ*ε* , we obtain the following equations:

$$\begin{split} 4\mathbf{C}^{2}\Phi\_{\mathbf{1}\tilde{\boldsymbol{\chi}}} &= \ -\mathsf{C}\Big(\Phi\_{0\boldsymbol{\chi}}^{(R)}\Phi\_{0\boldsymbol{\chi}}^{(L)} - \Phi\_{0\boldsymbol{\chi}}^{(R)}\Phi\_{0\tilde{\boldsymbol{\chi}}}^{(L)}\Big) \\ &+ \Big[\partial\_{\boldsymbol{\chi}}\Big(-2\mathsf{C}\Phi\_{0\boldsymbol{\Upsilon}}^{(R)} + \frac{\alpha}{4}\Phi\_{0\boldsymbol{\chi}\boldsymbol{\chi}}^{(R)} - \frac{3\mathsf{C}}{2}\Phi\_{0\boldsymbol{\chi}}^{(R)2}\Big) - \mathsf{C}^{2}\Phi\_{0\boldsymbol{\chi}\boldsymbol{\chi}}^{(R)}\Big] \\ &+ \Big[\partial\_{\tilde{\boldsymbol{\chi}}}\Big(2\mathsf{C}\Phi\_{0\boldsymbol{\Upsilon}}^{(L)} + \frac{\alpha}{4}\Phi\_{0\boldsymbol{\chi}\check{\boldsymbol{\chi}}\bar{\boldsymbol{\chi}}}^{(L)} + \frac{3\mathsf{C}}{2}\Phi\_{0\bar{\boldsymbol{\chi}}}^{(L)2}\Big) - \mathsf{C}^{2}\Phi\_{0\boldsymbol{\chi}\boldsymbol{\chi}}^{(L)}\Big], \end{split} \tag{38}$$

for the Cartesian geometry, and

$$\begin{split} 4\mathcal{C}^{2}\Phi\_{\mathrm{1\hat{q}\bar{q}}} &= -\mathsf{C} \Big( \Phi\_{0\mathrm{q}\bar{q}}^{(R)}\Phi\_{0\bar{q}}^{(L)} - \Phi\_{0\bar{q}}^{(R)}\Phi\_{0\bar{q}\bar{q}}^{(L)} \Big) \\ &+ \Big[ \partial\_{\rho} \Big( -2\mathsf{C}\Phi\_{0T}^{(R)} + \frac{\alpha}{4}\Phi\_{0\mathrm{qq}}^{(R)} - \frac{3\mathsf{C}}{2}\Phi\_{0\bar{q}}^{(R)2} - \frac{\mathsf{C}}{T}\Phi\_{0\bar{q}}^{(R)} \Big) - \frac{1}{T^{2}}\Phi\_{0\mathsf{q}\Theta}^{(R)} \Big] \\ &+ \Big[ \partial\_{\tilde{\boldsymbol{\eta}}} \Big( 2\mathsf{C}\Phi\_{0T}^{(L)} + \frac{\alpha}{4}\Phi\_{0\bar{q}\bar{q}}^{(L)} + \frac{3\mathsf{C}}{2}\Phi\_{0\bar{q}}^{(L)2} - \frac{\mathsf{C}}{T}\Phi\_{0\bar{\boldsymbol{\eta}}}^{(L)} \Big) - \frac{1}{T^{2}}\Phi\_{0\mathsf{q}\Theta}^{(R)} \Big), \end{split} \tag{39}$$

for the polar geometry. Here, it is important to observe the following. Once Eq. (38) is integrated in *χ* or ~*χ* [and, similarly, Eq. (39) in *ϱ* or *ϱ*~], secular terms arise: the secular therms are those in the square brackets in the right-hand side, because are functions of *χ* or ~*χ* (*ϱ* or ~*ϱ* in polar) alone, not both. Removal of these secular terms leads to two uncoupled nonlinear evolution equations for Φð Þ *<sup>R</sup>* <sup>0</sup> and <sup>Φ</sup>ð Þ *<sup>L</sup>* <sup>0</sup> in both geometries. Furthermore, using the equation Φ*<sup>T</sup>* ¼ �*n*<sup>1</sup> (i.e., the linear part of Eq. (29), together with Eqs. (26) and (27), it is found that the amplitude *ρ*<sup>1</sup> can also be decomposed to a left- and a right-going wave; this means that

$$
\rho\_1 = \rho\_1^{(\mathbb{R})} + \rho\_1^{(\mathbb{L})},
$$

with the fields *ρ* ð Þ *R* <sup>1</sup> and *ρ* ð Þ *L* <sup>1</sup> satisfying a pair of two uncoupled KP equations. In the Cartesian geometry, the KP equations are of the form:

$$\partial\_{\mathbb{X}} \left( \rho\_{1T}^{(\mathbb{R})} - \frac{a}{\mathbf{8C}} \rho\_{1\mathbb{X}\mathbb{X}}^{(\mathbb{R})} + \frac{\mathbf{3C}}{2} \rho\_{1}^{(\mathbb{R})} \rho\_{1\mathbb{X}}^{(\mathbb{R})} \right) + \frac{\mathbf{C}}{2} \rho\_{1\mathbb{X}\mathbb{Y}}^{(\mathbb{R})} = \mathbf{0},\tag{40}$$

$$
\partial\_{\ddot{\mathbb{X}}} \left( \rho\_{1T}^{(L)} + \frac{a}{8\mathsf{C}} \rho\_{1\ddot{\mathbb{X}}\ddot{\mathbb{X}}}^{(L)} - \frac{\mathsf{BC}}{2} \rho\_1^{(L)} \rho\_{1\ddot{\mathbb{X}}}^{(L)} \right) - \frac{\mathsf{C}}{2} \rho\_{1\dddot{\mathbb{X}}\ddot{\mathbb{Y}}}^{(L)} = \mathbf{0},\tag{41}
$$

while in the polar geometry, the relevant equations take the form of the so-called cylindrical KP (cKP), or Johnson's equations (see Refs. [22, 23]):

$$
\partial\_{\boldsymbol{\varrho}} \partial\_{\boldsymbol{\varrho}} \left( \rho\_{1\mathcal{T}}^{(\mathcal{R})} - \frac{a}{8\mathcal{C}} \rho\_{1\boldsymbol{\varrho}\boldsymbol{\varrho}\boldsymbol{\varrho}}^{(\mathcal{R})} + \frac{3\mathcal{C}}{2} \rho\_{1}^{(\mathcal{R})} \rho\_{1\boldsymbol{\varrho}}^{(\mathcal{R})} + \frac{1}{2\mathcal{T}} \rho\_{1}^{(\mathcal{R})} \right) + \frac{1}{2\mathcal{C}\mathcal{T}^{2}} \rho\_{1\boldsymbol{\Theta}\boldsymbol{\Theta}}^{(\mathcal{R})} = \mathbf{0},\tag{42}
$$

large (small) background amplitude or nonlocality corresponds to weak (strong)

We now employ the results of the above analysis and construct the solution of our original problem. This can be done as follows. Once a solution of Eq. (45) [or (46)] is known, an approximate solution of Eqs. (2) and (3) has the form:

It is important to note that Eq. (48) describes two different types of solitons, namely dark and antidark ones. Indeed, for *α*<0, i.e., for solutions satisfying the KPII equation (corresponding to the *strong nonlocality regime*) the solution (48) has the form of a *hump* on top of the cw background and is, thus, an *antidark soliton*. On

(corresponding to the *weak nonlocality regime*) the solution (48) has the form of a *dip* on top of the cw background and is, thus, a *dark soliton*. Note that in the local limit, with *d* ¼ 0, it turns out that *α* ¼ 1 and the solutions have always the form of dark solitons. In other words, antidark solitons are only supported by a sufficiently

Below, we will use known solutions of the KP and cKP equations and present corresponding solutions of the nonlocal NLS. In addition, we will perform direct numerical simulations to examine the evolution and – in some cases – the interac-

We start with the Cartesian geometry and consider, in particular, the simplest soliton solutions of the KP equation, the so-called line solitons. Specifically, the oneline soliton solution of Eq. (45), traveling at an angle to the *Y*-axis, reads [33]:

where *κ*, *λ* and *δ* are free parameters. We are particularly interested in the case where the corresponding solution of the NLS takes the form of an antidark soliton, with *U* obeying the KPII equation: since, in this case, line solitons are stable, we may expect that their interactions will give rise to patterns resembling those observed in shallow water [5]. An example of such an antidark soliton is ahown in **Figure 1**, for *t* ¼ 0. For this example, the following parameter values were used: *u*<sup>0</sup> ¼ 1 and *d* ¼

<sup>3</sup> <sup>p</sup> (leading to *<sup>α</sup>* ¼ �1*=*3<sup>&</sup>lt; 0), as well as *<sup>κ</sup>* <sup>¼</sup> *<sup>λ</sup>* <sup>¼</sup> 1, *<sup>δ</sup>* <sup>¼</sup> 0, and *<sup>ε</sup>* <sup>¼</sup> <sup>0</sup>*:*2. These

In what follows, we choose line solitons with specific parameters, so that their angle during interaction will determine the type of pattern. In the simulations, we evolve the initial configuration up to *t* ¼ 600, so that solitons have enough time to interact and generate interesting interaction patterns. In all cases, the numerical integration of Eqs. (2) and (3) is performed via a high-accuracy pseudo-spectral

sech<sup>2</sup>

ð Þ *Z* , *<sup>Z</sup>* � *<sup>κ</sup>* <sup>X</sup> <sup>þ</sup> *<sup>λ</sup>*<sup>Y</sup> � *<sup>κ</sup>*<sup>2</sup> <sup>þ</sup> <sup>3</sup>*λ*<sup>2</sup> � �<sup>T</sup> <sup>þ</sup> *<sup>δ</sup>* � �, (50)

*<sup>U</sup>*ð Þ¼ <sup>X</sup>,Y, <sup>T</sup> <sup>2</sup>*κ*<sup>2</sup>

2 *t* þ *i* 2 *αε*�1*=*<sup>2</sup> ðT 0 *U*dT <sup>0</sup> � �, (48)

*εαU:* (49)

exp �*i u*j j <sup>0</sup>

*n*≈ j j *u*<sup>0</sup>

the other hand, for *α* >0, i.e., for solutions satisfying the KPI equation

tion dynamics of the approximate soliton solutions (48).

**3.1 Antidark stripe solitons and their interactions**

surface tension, leading to KPII (KPI).

*Water Waves and Light: Two Unlikely Partners DOI: http://dx.doi.org/10.5772/intechopen.95431*

**3. Approximate soliton solutions**

*<sup>u</sup>* <sup>≈</sup>*u*<sup>0</sup> <sup>1</sup> � *<sup>ε</sup> <sup>α</sup>*

strong nonlocality.

1*=* ffiffiffi

**9**

values will also be used below.

2j j *u*<sup>0</sup> <sup>2</sup> *U*

!1*=*<sup>2</sup>

$$\partial\_{\bar{\boldsymbol{\varrho}}} \left( \rho\_{1T}^{(L)} + \frac{\boldsymbol{a}}{8\mathcal{C}} \rho\_{1\bar{\boldsymbol{\varrho}}\bar{\boldsymbol{\varrho}}\bar{\boldsymbol{\varrho}}}^{(L)} - \frac{3\mathcal{C}}{2} \rho\_1^{(L)} \rho\_{1\bar{\boldsymbol{\varrho}}}^{(L)} - \frac{1}{2T} \rho\_1^{(L)} \right) - \frac{1}{2\mathcal{C}T^2} \rho\_{1\bar{\boldsymbol{\Theta}}\bar{\boldsymbol{\Theta}}}^{(L)} = \mathbf{0}. \tag{43}$$

#### *2.2.3 Versions of the KP equations*

It is now convenient to further simplify the equations above, namely the KP and the cKP equations, in order to express them in their standard form [4]. We consider right-going waves in the Cartesian geometry, and radially expanding ones in the polar geometry, and introduce the following transformations,

$$\mathcal{Y} \mapsto \sqrt{\frac{3|a|}{4\mathcal{C}}} \mathcal{Y}, \quad \Theta \to \sqrt{\frac{3|a|}{4}} \Theta, \quad \mathcal{T} \mapsto -\frac{a}{8\mathcal{C}} \mathcal{T}, \quad \rho\_1^{(\mathbb{R})} = -\frac{a}{2\mathcal{C}^2} U,\tag{44}$$

which cast the respective KP and cKP equations into the following form:

$$
\partial\_{\lambda}(U\_T + \mathsf{G}UU\_{\lambda} + U\_{\lambda \mathcal{X}\mathcal{X}}) + \mathsf{3}\sigma^2 U\_{\mathcal{Y}\mathcal{Y}} = \mathbf{0},\tag{45}
$$

$$
\partial\_{\varrho} \left( U\_T + \mathfrak{G} U U\_{\varrho} + U\_{\varrho\varrho\varrho} + \frac{1}{2T} U \right) + \frac{3\sigma^2}{T^2} U\_{\varrho\Theta} = 0. \tag{46}
$$

Here, the parameter *σ*<sup>2</sup> is defined as:

$$
\sigma^2 = -\operatorname{sgn} a = \operatorname{sgn}\left(4d|u\_0|^2 - 1\right). \tag{47}
$$

It is now clear that, in the Cartesian case, Eq. (45) includes both versions of the KP equation, KPI and KPII [33]. Indeed, for *<sup>σ</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> ) *<sup>α</sup>* <sup>&</sup>lt;0, i.e., for j j *<sup>u</sup>*<sup>0</sup> <sup>2</sup> <sup>&</sup>gt; <sup>1</sup>*=*ð Þ <sup>4</sup>*<sup>d</sup>* or *<sup>d</sup>*>1*=*ð Þ <sup>4</sup>j*u*0<sup>j</sup> <sup>2</sup> , Eq. (45) is a KPII equation; this corresponds to a small effective surface tension. On the other hand, for *<sup>σ</sup>*<sup>2</sup> ¼ �<sup>1</sup> ) *<sup>α</sup>* <sup>&</sup>gt;0, i.e., j j *<sup>u</sup>*<sup>0</sup> <sup>2</sup> <sup>&</sup>lt;1*=*ð Þ <sup>4</sup>*<sup>d</sup>* or *<sup>d</sup>*<1*=*ð Þ <sup>4</sup>j*u*0<sup>j</sup> <sup>2</sup> , Eq. (45) is a KPI equation; this corresponds to the case of large effective surface tension (see end of Section 2.1 and discussion below).

In other words, for a fixed degree of nonlocality *d*, a larger (smaller) background amplitude ∣*u*0∣, as defined by the sign of *α*, corresponds to KPII (KPI); similarly, for a fixed background amplitude ∣*u*0∣, a strong (weak) nonlocality, as defined by the above regimes of *d*, corresponds to KPII (KPI). Notice that in the local limit of *d* ¼ 0, the asymptotic analysis leads only to the KPI model, in which line solitons are unstable; this fact was used to better understand self-focusing and transverse instability of plane dark solitons of the defocusing NLS Equation [19, 20]. Notably, the same parameter *α* can also been shown to distinguish solutions in 1D [34] and radially symmetric [35] systems; a similar case, pertinent to the polar geometry, will be considered below.

Once again, it is important to highlight the fact that the existence of these regimes resembles the situation occurring in shallow water. In this context, weak surface tension corresponds to *<sup>σ</sup>*<sup>2</sup> <sup>¼</sup> 1 in Eq. (45) (i.e., KPII), while strong surface tension is pertinent to *<sup>σ</sup>*<sup>2</sup> ¼ �1 (i.e., KPI). Thus, there exists an immediate connection between the original problem with the one of shallow water waves: relatively

large (small) background amplitude or nonlocality corresponds to weak (strong) surface tension, leading to KPII (KPI).

## **3. Approximate soliton solutions**

while in the polar geometry, the relevant equations take the form of the so-called

1 <sup>2</sup><sup>T</sup> *<sup>ρ</sup>* ð Þ *R* 1

It is now convenient to further simplify the equations above, namely the KP and the cKP equations, in order to express them in their standard form [4]. We consider right-going waves in the Cartesian geometry, and radially expanding ones in the

<sup>Θ</sup>, <sup>T</sup> <sup>↦</sup> � *<sup>α</sup>*

which cast the respective KP and cKP equations into the following form:

1 <sup>2</sup><sup>T</sup> *<sup>U</sup>*

It is now clear that, in the Cartesian case, Eq. (45) includes both versions of the

, Eq. (45) is a KPII equation; this corresponds to a small effective

, Eq. (45) is a KPI equation; this corresponds to the case of large

In other words, for a fixed degree of nonlocality *d*, a larger (smaller) background amplitude ∣*u*0∣, as defined by the sign of *α*, corresponds to KPII (KPI); similarly, for a fixed background amplitude ∣*u*0∣, a strong (weak) nonlocality, as defined by the above regimes of *d*, corresponds to KPII (KPI). Notice that in the local limit of *d* ¼ 0, the asymptotic analysis leads only to the KPI model, in which line solitons are unstable; this fact was used to better understand self-focusing and transverse instability of plane dark solitons of the defocusing NLS Equation [19, 20]. Notably, the same parameter *α* can also been shown to distinguish solutions in 1D [34] and radially symmetric [35] systems; a similar case, pertinent to the polar geometry, will

Once again, it is important to highlight the fact that the existence of these regimes resembles the situation occurring in shallow water. In this context, weak surface tension corresponds to *<sup>σ</sup>*<sup>2</sup> <sup>¼</sup> 1 in Eq. (45) (i.e., KPII), while strong surface tension is pertinent to *<sup>σ</sup>*<sup>2</sup> ¼ �1 (i.e., KPI). Thus, there exists an immediate connection between the original problem with the one of shallow water waves: relatively

*<sup>∂</sup>*<sup>X</sup> <sup>ð</sup>*U*<sup>T</sup> <sup>þ</sup> <sup>6</sup>*UU*<sup>X</sup> <sup>þ</sup> *<sup>U</sup>*XXX Þ þ <sup>3</sup>*σ*<sup>2</sup>

*<sup>σ</sup>*<sup>2</sup> ¼ � sgn *<sup>α</sup>* <sup>¼</sup> sgn 4*d u*j j <sup>0</sup>

KP equation, KPI and KPII [33]. Indeed, for *<sup>σ</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> ) *<sup>α</sup>* <sup>&</sup>lt;0, i.e., for j j *<sup>u</sup>*<sup>0</sup>

effective surface tension (see end of Section 2.1 and discussion below).

surface tension. On the other hand, for *<sup>σ</sup>*<sup>2</sup> ¼ �<sup>1</sup> ) *<sup>α</sup>* <sup>&</sup>gt;0, i.e., j j *<sup>u</sup>*<sup>0</sup>

� �

þ

<sup>8</sup>*C*<sup>T</sup> , *<sup>ρ</sup>*

þ 3*σ*<sup>2</sup>

<sup>2</sup> � <sup>1</sup> � �

ð Þ *R* <sup>1</sup> ¼ � *<sup>α</sup>*

� <sup>1</sup> <sup>2</sup>*C*<sup>T</sup> <sup>2</sup> *<sup>ρ</sup>*

1 <sup>2</sup>*C*<sup>T</sup> <sup>2</sup> *<sup>ρ</sup>*

ð Þ *R*

ð Þ *L*

<sup>1</sup>ΘΘ ¼ 0, (42)

<sup>1</sup>ΘΘ ¼ 0*:* (43)

<sup>2</sup>*C*<sup>2</sup> *<sup>U</sup>*, (44)

*U*YY ¼ 0, (45)

<sup>T</sup> <sup>2</sup> *<sup>U</sup>*ΘΘ <sup>¼</sup> <sup>0</sup>*:* (46)

*:* (47)

<sup>2</sup> <sup>&</sup>lt;1*=*ð Þ <sup>4</sup>*<sup>d</sup>* or

<sup>2</sup> <sup>&</sup>gt; <sup>1</sup>*=*ð Þ <sup>4</sup>*<sup>d</sup>* or

cylindrical KP (cKP), or Johnson's equations (see Refs. [22, 23]):

3*C* 2 *ρ* ð Þ *R* <sup>1</sup> *ρ* ð Þ *R* 1*<sup>χ</sup>* þ

� �

� �

polar geometry, and introduce the following transformations,

ffiffiffiffiffiffiffiffi 3∣*α*∣ 4

*<sup>∂</sup><sup>ϱ</sup> <sup>U</sup>*<sup>T</sup> <sup>þ</sup> <sup>6</sup>*UU<sup>ϱ</sup>* <sup>þ</sup> *<sup>U</sup>*ϱϱϱ <sup>þ</sup>

r

*∂<sup>ϱ</sup> ρ* ð Þ *R* <sup>1</sup><sup>T</sup> � *<sup>α</sup>* <sup>8</sup>*C<sup>ρ</sup>* ð Þ *R* 1ϱϱϱ þ

*∂<sup>ϱ</sup>*<sup>~</sup> *ρ* ð Þ *L* <sup>1</sup><sup>T</sup> <sup>þ</sup> *<sup>α</sup>* <sup>8</sup>*C<sup>ρ</sup>* ð Þ *L* <sup>1</sup>*ϱ*~~*ϱ*~*<sup>ϱ</sup>* � <sup>3</sup>*<sup>C</sup>* 2 *ρ* ð Þ *L* <sup>1</sup> *ρ* ð Þ *L* <sup>1</sup>~*<sup>ϱ</sup>* � <sup>1</sup> <sup>2</sup><sup>T</sup> *<sup>ρ</sup>* ð Þ *L* 1

Y ↦

*<sup>d</sup>*>1*=*ð Þ <sup>4</sup>j*u*0<sup>j</sup> <sup>2</sup>

*<sup>d</sup>*<1*=*ð Þ <sup>4</sup>j*u*0<sup>j</sup> <sup>2</sup>

be considered below.

**8**

*2.2.3 Versions of the KP equations*

*Nonlinear Optics - From Solitons to Similaritons*

ffiffiffiffiffiffiffiffi 3∣*α*∣ 4*C*<sup>2</sup>

Y, Θ !

Here, the parameter *σ*<sup>2</sup> is defined as:

r

We now employ the results of the above analysis and construct the solution of our original problem. This can be done as follows. Once a solution of Eq. (45) [or (46)] is known, an approximate solution of Eqs. (2) and (3) has the form:

$$u \approx u\_0 \left(1 - \varepsilon \frac{a}{2|u\_0|^2} U\right)^{1/2} \exp\left(-i|u\_0|^2 t + \frac{i}{2} a e^{-1/2} \int\_0^T U d\mathcal{T}'\right),\tag{48}$$

$$n \approx \left| u\_0 \right|^2 - \frac{1}{2} \epsilon a U. \tag{49}$$

It is important to note that Eq. (48) describes two different types of solitons, namely dark and antidark ones. Indeed, for *α*<0, i.e., for solutions satisfying the KPII equation (corresponding to the *strong nonlocality regime*) the solution (48) has the form of a *hump* on top of the cw background and is, thus, an *antidark soliton*. On the other hand, for *α* >0, i.e., for solutions satisfying the KPI equation (corresponding to the *weak nonlocality regime*) the solution (48) has the form of a *dip* on top of the cw background and is, thus, a *dark soliton*. Note that in the local limit, with *d* ¼ 0, it turns out that *α* ¼ 1 and the solutions have always the form of dark solitons. In other words, antidark solitons are only supported by a sufficiently strong nonlocality.

Below, we will use known solutions of the KP and cKP equations and present corresponding solutions of the nonlocal NLS. In addition, we will perform direct numerical simulations to examine the evolution and – in some cases – the interaction dynamics of the approximate soliton solutions (48).

#### **3.1 Antidark stripe solitons and their interactions**

We start with the Cartesian geometry and consider, in particular, the simplest soliton solutions of the KP equation, the so-called line solitons. Specifically, the oneline soliton solution of Eq. (45), traveling at an angle to the *Y*-axis, reads [33]:

$$\begin{aligned} U(\mathcal{X}, \mathcal{Y}, \mathcal{T}) &= 2\kappa^2 \text{sech}^2(\mathcal{Z}), \\ \mathcal{Z} \equiv \kappa \left[ \mathcal{X} + \lambda \mathcal{Y} - \left( \kappa^2 + 3\lambda^2 \right) \mathcal{T} + \delta \right], \end{aligned} \tag{50}$$

where *κ*, *λ* and *δ* are free parameters. We are particularly interested in the case where the corresponding solution of the NLS takes the form of an antidark soliton, with *U* obeying the KPII equation: since, in this case, line solitons are stable, we may expect that their interactions will give rise to patterns resembling those observed in shallow water [5]. An example of such an antidark soliton is ahown in **Figure 1**, for *t* ¼ 0. For this example, the following parameter values were used: *u*<sup>0</sup> ¼ 1 and *d* ¼ 1*=* ffiffiffi <sup>3</sup> <sup>p</sup> (leading to *<sup>α</sup>* ¼ �1*=*3<sup>&</sup>lt; 0), as well as *<sup>κ</sup>* <sup>¼</sup> *<sup>λ</sup>* <sup>¼</sup> 1, *<sup>δ</sup>* <sup>¼</sup> 0, and *<sup>ε</sup>* <sup>¼</sup> <sup>0</sup>*:*2. These values will also be used below.

In what follows, we choose line solitons with specific parameters, so that their angle during interaction will determine the type of pattern. In the simulations, we evolve the initial configuration up to *t* ¼ 600, so that solitons have enough time to interact and generate interesting interaction patterns. In all cases, the numerical integration of Eqs. (2) and (3) is performed via a high-accuracy pseudo-spectral

Miles [7, 8]). This feature is clearly observed in the figure. Parameter values in this case are: *κ*<sup>1</sup> ¼ *κ*2*=*2 ¼ 1*=*2, *λ*<sup>1</sup> ¼ 3*λ*<sup>2</sup> ¼ 3*=*4 (and for all cases, also pertaining to X-type

Next, we proceed with the X-type interactions, which can be discriminated according to the resulting "stem": an interaction with a *short stem* (**Figure 3**-left), an interaction with a long stem where the stem height is higher than the incoming line

In the same category falls an X-type interaction with a long stem, with the stem height being lower than the tallest incoming line soliton, which we refer to here as an H-interaction. A typical such interaction is given in **Figure 4**, and parameter

An immediate generalization of the above is a three-wave interaction, where

*<sup>η</sup>i*þ*<sup>η</sup> <sup>j</sup>*þ*Aij* <sup>þ</sup> *<sup>e</sup>*

� �<sup>2</sup> � *<sup>λ</sup><sup>i</sup>* � *<sup>λ</sup> <sup>j</sup>*

� �<sup>2</sup> � *<sup>λ</sup><sup>i</sup>* � *<sup>λ</sup> <sup>j</sup>*

� �<sup>2</sup>

*<sup>η</sup>*1þ*η*2þ*η*3þ*A*12þ*A*13þ*A*<sup>23</sup> , (52)

� �<sup>2</sup> , (53)

*e*

*κ<sup>i</sup>* þ *κ <sup>j</sup>*

interactions below, *δ*<sup>1</sup> ¼ *δ*<sup>2</sup> ¼ 0).

*Water Waves and Light: Two Unlikely Partners DOI: http://dx.doi.org/10.5772/intechopen.95431*

now

and

**Figure 3.**

**Figure 4.**

**11**

solitons (**Figure 3**-middle and right respectively).

values: *<sup>κ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>κ</sup>*2*=*<sup>2</sup> <sup>¼</sup> <sup>1</sup>*=*2, *<sup>λ</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup>*=*<sup>2</sup> � <sup>10</sup>�7, and *<sup>λ</sup>*<sup>2</sup> <sup>¼</sup> 0.

*<sup>η</sup><sup>i</sup>* <sup>þ</sup> <sup>X</sup> 1≤*i*<*j*≤3

exp *Aij* � � <sup>¼</sup> *<sup>κ</sup><sup>i</sup>* � *<sup>κ</sup> <sup>j</sup>*

so that the resulting interaction is as in **Figure 5**, for *t* ¼ 200. Here, *κ*<sup>1</sup> ¼ 1, *κ*<sup>2</sup> ¼ 2, *κ*<sup>3</sup> ¼ 3, *λ*<sup>1</sup> ¼ *λ*2*=*2 ¼ *λ*3*=*5 ¼ �1*=*3 and, as before, *δ*<sup>1</sup> ¼ *δ*<sup>2</sup> ¼ *δ*<sup>3</sup> ¼ 0.

*X-type interactions, u x*ð Þ , *y*, 0 *, with: κ*<sup>1</sup> ¼ *κ*<sup>2</sup> ¼ 1*=*2*, λ*<sup>1</sup> ¼ �*λ*<sup>2</sup> ¼ �2*=*3 *(left), κ*<sup>1</sup> ¼ *κ*<sup>2</sup> ¼ 1*=*2*, λ*<sup>1</sup> ¼ <sup>1</sup>*=*<sup>4</sup> � <sup>0</sup>*:*01*, <sup>λ</sup>*<sup>2</sup> <sup>¼</sup> <sup>3</sup>*=*<sup>4</sup> *(middle), <sup>κ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>κ</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup>*=*2*, <sup>λ</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup>*=*2*, <sup>λ</sup>*<sup>2</sup> ¼ �1*=*<sup>2</sup> � <sup>10</sup>�<sup>10</sup> *(right). Observe that the*

*An H-type interaction, u x*ð Þ , *<sup>y</sup>*, 0 *, with <sup>κ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>κ</sup>*2*=*<sup>2</sup> <sup>¼</sup> <sup>1</sup>*=*2*, <sup>λ</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup>*=*<sup>2</sup> � <sup>10</sup>�<sup>7</sup>*, <sup>λ</sup>*<sup>2</sup> <sup>¼</sup> <sup>0</sup>*.*

1≤*i*≤3 *e*

*length of the stem is increased from left to right (see also text).*

*<sup>F</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>X</sup>

**Figure 1.**

*A typical single antidark soliton solution of the nonlocal NLS (u x*ð Þ , *y*, 0 *), with κ* ¼ *λ* ¼ 1*, δ* ¼ 0*; other parameter values are: u*<sup>0</sup> ¼ *C* ¼ 1*, d* ¼ 1*=* ffiffiffi <sup>3</sup> <sup>p</sup> *, and <sup>ε</sup>* <sup>¼</sup> <sup>0</sup>*:*2*.*

method in space and fourth-order in time. To adjust to the periodic boundary conditions, the scheme requires we place the solutions on top of a wide super-Gaussian background that decays at infinity and makes the initial condition periodic. As it will be seen below, interacting antidark NLS solitons follow closely the KP dynamics, giving rise to patterns that are usually observed in water.

We start with the interaction of two solitons, and express the two-line soliton solution of KPII as follows:

$$\begin{aligned} U(\mathcal{X}, \mathcal{Y}, T) &= 2\partial\_{\mathcal{X}}^2 \ln F(\mathcal{X}, \mathcal{Y}, T), \\ F \equiv \mathbf{1} + \exp\left(\mathbf{Z}\_1\right) + \exp\left(\mathbf{Z}\_2\right) + \exp\left(\mathbf{Z}\_1 + \mathbf{Z}\_2 + \mathbf{A}\_{12}\right), \\ \exp\left(\mathbf{A}\_{12}\right) &= \frac{\left(\kappa\_1 - \kappa\_2\right)^2 - \left(\lambda\_1 - \lambda\_2\right)^2}{\left(\kappa\_1 + \kappa\_2\right)^2 - \left(\lambda\_1 - \lambda\_2\right)^2}, \end{aligned} \tag{51}$$

with *Zi* � *<sup>κ</sup><sup>i</sup>* <sup>X</sup> <sup>þ</sup> *<sup>λ</sup>i*<sup>Y</sup> � *<sup>κ</sup>*<sup>2</sup> *<sup>i</sup>* <sup>þ</sup> <sup>3</sup>*λ*<sup>2</sup> *i* � �<sup>T</sup> <sup>þ</sup> *<sup>δ</sup><sup>i</sup>* � �.

We first focus on interacting line solitons that form Y- and X-shaped waveforms, and proceed with more complex ones later in the text. In **Figure 2**, we show a typical Y-shaped pattern, whereby the resonant two-soliton interaction gives rise to the emergence of one soliton, with its maximum height being four times that of the incoming solitons (this resonant Y-shaped solution was originally found by

#### **Figure 2.**

*A typical Y-type interaction, u x*ð Þ , *y*, 0 *, whereby out of two incoming colliding solitons, a single line soliton emerges. Parameter values are: κ*<sup>1</sup> ¼ *κ*2*=*2 ¼ 1*=*2*, λ*<sup>1</sup> ¼ 3*λ*<sup>2</sup> ¼ 3*=*4*.*

*Water Waves and Light: Two Unlikely Partners DOI: http://dx.doi.org/10.5772/intechopen.95431*

Miles [7, 8]). This feature is clearly observed in the figure. Parameter values in this case are: *κ*<sup>1</sup> ¼ *κ*2*=*2 ¼ 1*=*2, *λ*<sup>1</sup> ¼ 3*λ*<sup>2</sup> ¼ 3*=*4 (and for all cases, also pertaining to X-type interactions below, *δ*<sup>1</sup> ¼ *δ*<sup>2</sup> ¼ 0).

Next, we proceed with the X-type interactions, which can be discriminated according to the resulting "stem": an interaction with a *short stem* (**Figure 3**-left), an interaction with a long stem where the stem height is higher than the incoming line solitons (**Figure 3**-middle and right respectively).

In the same category falls an X-type interaction with a long stem, with the stem height being lower than the tallest incoming line soliton, which we refer to here as an H-interaction. A typical such interaction is given in **Figure 4**, and parameter values: *<sup>κ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>κ</sup>*2*=*<sup>2</sup> <sup>¼</sup> <sup>1</sup>*=*2, *<sup>λ</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup>*=*<sup>2</sup> � <sup>10</sup>�7, and *<sup>λ</sup>*<sup>2</sup> <sup>¼</sup> 0.

An immediate generalization of the above is a three-wave interaction, where now

$$F = \mathbf{1} + \sum\_{1 \le i \le 3} \mathbf{e}^{\eta\_i} + \sum\_{1 \le i < j \le 3} \mathbf{e}^{\eta\_i + \eta\_j + A\_{\bar{\eta}}} + \mathbf{e}^{\eta\_1 + \eta\_2 + \eta\_3 + A\_{12} + A\_{13} + A\_{23}},\tag{52}$$

and

method in space and fourth-order in time. To adjust to the periodic boundary conditions, the scheme requires we place the solutions on top of a wide super-Gaussian background that decays at infinity and makes the initial condition periodic. As it will be seen below, interacting antidark NLS solitons follow closely the

*A typical single antidark soliton solution of the nonlocal NLS (u x*ð Þ , *y*, 0 *), with κ* ¼ *λ* ¼ 1*, δ* ¼ 0*; other*

<sup>3</sup> <sup>p</sup> *, and <sup>ε</sup>* <sup>¼</sup> <sup>0</sup>*:*2*.*

We start with the interaction of two solitons, and express the two-line soliton

*F* � 1 þ exp ð Þþ *Z*<sup>1</sup> exp ð Þþ *Z*<sup>2</sup> exp ð Þ *Z*<sup>1</sup> þ *Z*<sup>2</sup> þ *A*<sup>12</sup> ,

ð Þ *κ*<sup>1</sup> � *κ*<sup>2</sup>

ð Þ *κ*<sup>1</sup> þ *κ*<sup>2</sup>

We first focus on interacting line solitons that form Y- and X-shaped waveforms, and proceed with more complex ones later in the text. In **Figure 2**, we show a typical Y-shaped pattern, whereby the resonant two-soliton interaction gives rise to the emergence of one soliton, with its maximum height being four times that of the incoming solitons (this resonant Y-shaped solution was originally found by

*A typical Y-type interaction, u x*ð Þ , *y*, 0 *, whereby out of two incoming colliding solitons, a single line soliton*

*emerges. Parameter values are: κ*<sup>1</sup> ¼ *κ*2*=*2 ¼ 1*=*2*, λ*<sup>1</sup> ¼ 3*λ*<sup>2</sup> ¼ 3*=*4*.*

<sup>X</sup> ln *F*ð Þ X, Y, T ,

<sup>2</sup> � ð Þ *<sup>λ</sup>*<sup>1</sup> � *<sup>λ</sup>*<sup>2</sup>

<sup>2</sup> � ð Þ *<sup>λ</sup>*<sup>1</sup> � *<sup>λ</sup>*<sup>2</sup>

2

2 ,

(51)

KP dynamics, giving rise to patterns that are usually observed in water.

*<sup>U</sup>*ð Þ¼ <sup>X</sup>, <sup>Y</sup>, <sup>T</sup> <sup>2</sup>*∂*<sup>2</sup>

exp ð Þ¼ *A*<sup>12</sup>

� �.

*<sup>i</sup>* <sup>þ</sup> <sup>3</sup>*λ*<sup>2</sup> *i* � �<sup>T</sup> <sup>þ</sup> *<sup>δ</sup><sup>i</sup>*

solution of KPII as follows:

*parameter values are: u*<sup>0</sup> ¼ *C* ¼ 1*, d* ¼ 1*=* ffiffiffi

*Nonlinear Optics - From Solitons to Similaritons*

**Figure 1.**

**Figure 2.**

**10**

with *Zi* � *<sup>κ</sup><sup>i</sup>* <sup>X</sup> <sup>þ</sup> *<sup>λ</sup>i*<sup>Y</sup> � *<sup>κ</sup>*<sup>2</sup>

$$\exp\left(A\_{i\bar{j}}\right) = \frac{\left(\kappa\_i - \kappa\_j\right)^2 - \left(\lambda\_i - \lambda\_j\right)^2}{\left(\kappa\_i + \kappa\_j\right)^2 - \left(\lambda\_i - \lambda\_j\right)^2},\tag{53}$$

so that the resulting interaction is as in **Figure 5**, for *t* ¼ 200. Here, *κ*<sup>1</sup> ¼ 1, *κ*<sup>2</sup> ¼ 2, *κ*<sup>3</sup> ¼ 3, *λ*<sup>1</sup> ¼ *λ*2*=*2 ¼ *λ*3*=*5 ¼ �1*=*3 and, as before, *δ*<sup>1</sup> ¼ *δ*<sup>2</sup> ¼ *δ*<sup>3</sup> ¼ 0.

#### **Figure 3.**

*X-type interactions, u x*ð Þ , *y*, 0 *, with: κ*<sup>1</sup> ¼ *κ*<sup>2</sup> ¼ 1*=*2*, λ*<sup>1</sup> ¼ �*λ*<sup>2</sup> ¼ �2*=*3 *(left), κ*<sup>1</sup> ¼ *κ*<sup>2</sup> ¼ 1*=*2*, λ*<sup>1</sup> ¼ <sup>1</sup>*=*<sup>4</sup> � <sup>0</sup>*:*01*, <sup>λ</sup>*<sup>2</sup> <sup>¼</sup> <sup>3</sup>*=*<sup>4</sup> *(middle), <sup>κ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>κ</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup>*=*2*, <sup>λ</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup>*=*2*, <sup>λ</sup>*<sup>2</sup> ¼ �1*=*<sup>2</sup> � <sup>10</sup>�<sup>10</sup> *(right). Observe that the length of the stem is increased from left to right (see also text).*

**Figure 4.** *An H-type interaction, u x*ð Þ , *<sup>y</sup>*, 0 *, with <sup>κ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>κ</sup>*2*=*<sup>2</sup> <sup>¼</sup> <sup>1</sup>*=*2*, <sup>λ</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup>*=*<sup>2</sup> � <sup>10</sup>�<sup>7</sup>*, <sup>λ</sup>*<sup>2</sup> <sup>¼</sup> <sup>0</sup>*.*

**Figure 5.** *A 2–3 wave interaction, u x*ð Þ , *y*, 0 *, with κ*<sup>1</sup> ¼ 1*, κ*<sup>2</sup> ¼ 2*, κ*<sup>3</sup> ¼ 3*, λ*<sup>1</sup> ¼ *λ*2*=*2 ¼ *λ*3*=*5 ¼ �1*=*3 *and δ*<sup>1</sup> ¼ *δ*<sup>2</sup> ¼ *δ*<sup>3</sup> ¼ 0*.*

More intricate structures may still form whose mathematical description is more complex. Such novel weblike structures for the KP equation, termed *N*-in *M*-out, have been found using Wronskian methods [11, 13, 15] and the so-called *τ*-function. We illustrate the basic construction of these solitons but do not go into further detail as this is beyond the scope of this work. The most convenient way to write the line soliton solutions of the KPII system is through the Wronskian, or *τ*-function, which replaces the *F* function above. Now, the solutions of the KPII equation are written in a similar manner as above, as

$$U(\mathcal{X}, \mathcal{Y}, \mathcal{T}) = 2\partial\_{\mathcal{X}}^2 \ln \tau(\mathcal{X}, \mathcal{Y}, \mathcal{T}),\tag{54}$$

We start with a 2–2 interaction, by defining the matrix *A*, such that

and the relative real parameters *k*<sup>1</sup> ¼ �1, *k*<sup>2</sup> ¼ 0, *k*<sup>3</sup> ¼ 1 and *k*<sup>4</sup> ¼ 2

4

Similarly, for a 3–3 wave interaction one has:

*A* ¼

following interaction pattern of **Figure 7**.

matrices are called totally non-negative matrices.

[36] for a stability analysis of KP line solitons.

0

B@

corresponding to the relative *θm*j

depicted in **Figure 6**.

**Figure 6.**

**Figure 7.**

*variable u x*ð Þ , *y*, *t .*

*variable u x*ð Þ , *y*, *t .*

*Water Waves and Light: Two Unlikely Partners DOI: http://dx.doi.org/10.5772/intechopen.95431*

**3.2 Dark lump solitons**

**13**

*<sup>A</sup>* <sup>¼</sup> 1 0 �<sup>1</sup> �<sup>1</sup>

*A typical 3–3 wave evolution, for t* ¼ 0 *(left), t* ¼ 400 *(middle), and t* ¼ 600 *(right). All figures refer to the*

*A typical 2–2 wave evolution for t* ¼ 0 *(left), t* ¼ 200 *(middle) and t* ¼ 400 *(right). All figures refer to the*

1 0 �1 �10 2 01 2 1 0 �1 00 0 0 1 1

and *k*<sup>1</sup> ¼ �1, *k*<sup>2</sup> ¼ �1*=*2, *k*<sup>3</sup> ¼ 0, *k*<sup>4</sup> ¼ 1*=*2, *k*<sup>5</sup> ¼ 1 and *k*<sup>6</sup> ¼ 3*=*2 to produce the

It is important to note here that the matrices *A* for the 2–2 and 3–3 solitons, are not arbitrarily chosen. The *N* � *N* maximal minors of *A* must be non-negative in order for the KP solution *q* to be regular for all *X*, *Y* and *T*. In the literature, such

So far, we have focused on structures obeying the effective KPII equation. These

structures, namely the antidark stripe solitons, are quasi one-dimensional (1D) states, which are stable. The respective states that satisfy the effective KPI equation, namely dark stripe solitons, are expected to be transversely unstable – see, e.g., Ref.

01 1 1 � �, (58)

*<sup>m</sup>*¼1. Then, the evolution of this interaction is

1

CA, (59)

where this new *τ*-function represents the Wronskian determinant

$$
\pi(\mathcal{X}, \mathcal{Y}, T) = \begin{pmatrix}
f\_1 & f\_2 & \cdots & f\_N \\
f\_1' & f\_2' & \cdots & f\_N' \\
\vdots & \vdots & \cdots & \vdots \\
f\_1^{(N-1)} & f\_2^{(N-1)} & \cdots & f\_N^{(N-1)}
\end{pmatrix}.\tag{55}$$

Here superscripts denote differentiation with respect to *X*, and the set of *f <sup>n</sup>* functions constitute the set of linearly independent solutions of the system:

$$f\_{\mathcal{Y}} = f\_{\mathcal{AX}}, \quad f\_{\mathcal{T}} = f\_{\mathcal{AX}\mathcal{X}}.\tag{56}$$

In particular, for line solitons these are defined as:

$$f\_n(\mathcal{X}, \mathcal{Y}, T) = \sum\_{m=1}^{M} a\_{nm} e^{\theta\_m}, \ n = 1, 2, \dots, N,\tag{57}$$

where *<sup>θ</sup><sup>m</sup>* <sup>¼</sup> *km*<sup>X</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> *<sup>m</sup>*<sup>Y</sup> <sup>þ</sup> *<sup>k</sup>*<sup>3</sup> *<sup>m</sup>*T þ *θ*0*<sup>m</sup>* with distinct real parameters *km* 's with the property: *k*<sup>1</sup> <*k*<sup>2</sup> < ⋯ <*kM*. The parameters *θ*0*<sup>m</sup>* are real constants. Importantly, the coefficients *anm* define an *N* � *M* matrix of rank *N*, due to the linear independence of the functions *f <sup>n</sup>*, such that ð Þ *anm*≕*A* . Below we provide two examples, and their evolution for a 2–2 and 3–3 interaction (see **Figures 6** and **7**, respectively).

*Water Waves and Light: Two Unlikely Partners DOI: http://dx.doi.org/10.5772/intechopen.95431*

**Figure 6.**

*A typical 2–2 wave evolution for t* ¼ 0 *(left), t* ¼ 200 *(middle) and t* ¼ 400 *(right). All figures refer to the variable u x*ð Þ , *y*, *t .*

**Figure 7.**

More intricate structures may still form whose mathematical description is more complex. Such novel weblike structures for the KP equation, termed *N*-in *M*-out, have been found using Wronskian methods [11, 13, 15] and the so-called *τ*-function. We illustrate the basic construction of these solitons but do not go into further detail as this is beyond the scope of this work. The most convenient way to write the line soliton solutions of the KPII system is through the Wronskian, or *τ*-function, which replaces the *F* function above. Now, the solutions of the KPII equation are

*f* <sup>1</sup> *f* <sup>2</sup> ⋯ *f <sup>N</sup>*

⋮ ⋮⋯⋮

ð Þ *N*�1 <sup>2</sup> ⋯ *f*

Here superscripts denote differentiation with respect to *X*, and the set of *f <sup>n</sup>* functions constitute the set of linearly independent solutions of the system:

*anme*

the property: *k*<sup>1</sup> <*k*<sup>2</sup> < ⋯ <*kM*. The parameters *θ*0*<sup>m</sup>* are real constants. Importantly, the coefficients *anm* define an *N* � *M* matrix of rank *N*, due to the linear independence of the functions *f <sup>n</sup>*, such that ð Þ *anm*≕*A* . Below we provide two examples, and their evolution for a 2–2 and 3–3 interaction (see **Figures 6** and **7**, respectively).

<sup>2</sup> ⋯ *f*

<sup>X</sup> ln *τ*ð Þ X, Y, T , (54)

1

CCCCCCA

*<sup>θ</sup><sup>m</sup>* , *<sup>n</sup>* <sup>¼</sup> 1, 2, … , *<sup>N</sup>*, (57)

*:* (55)

0 *N*

ð Þ *N*�1 *N*

*f* <sup>Y</sup> ¼ *f* XX , *f* <sup>T</sup> ¼ *f* XXX *:* (56)

*<sup>m</sup>*T þ *θ*0*<sup>m</sup>* with distinct real parameters *km* 's with

*<sup>U</sup>*ð Þ¼ <sup>X</sup>,Y, <sup>T</sup> <sup>2</sup>*∂*<sup>2</sup>

*A 2–3 wave interaction, u x*ð Þ , *y*, 0 *, with κ*<sup>1</sup> ¼ 1*, κ*<sup>2</sup> ¼ 2*, κ*<sup>3</sup> ¼ 3*, λ*<sup>1</sup> ¼ *λ*2*=*2 ¼ *λ*3*=*5 ¼ �1*=*3

0

BBBBBB@

where this new *τ*-function represents the Wronskian determinant

*f* 0 <sup>1</sup> *f* 0

*f* ð Þ *N*�1 <sup>1</sup> *f*

*M*

*m*¼1

written in a similar manner as above, as

*Nonlinear Optics - From Solitons to Similaritons*

**Figure 5.**

*and δ*<sup>1</sup> ¼ *δ*<sup>2</sup> ¼ *δ*<sup>3</sup> ¼ 0*.*

*τ*ð Þ¼ X, Y, T

In particular, for line solitons these are defined as:

*<sup>f</sup> <sup>n</sup>*ð Þ¼ <sup>X</sup>,Y, <sup>T</sup> <sup>X</sup>

*<sup>m</sup>*<sup>Y</sup> <sup>þ</sup> *<sup>k</sup>*<sup>3</sup>

where *<sup>θ</sup><sup>m</sup>* <sup>¼</sup> *km*<sup>X</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup>

**12**

*A typical 3–3 wave evolution, for t* ¼ 0 *(left), t* ¼ 400 *(middle), and t* ¼ 600 *(right). All figures refer to the variable u x*ð Þ , *y*, *t .*

We start with a 2–2 interaction, by defining the matrix *A*, such that

$$A = \begin{pmatrix} 1 & 0 & -1 & -1 \\ 0 & 1 & 1 & 1 \end{pmatrix},\tag{58}$$

and the relative real parameters *k*<sup>1</sup> ¼ �1, *k*<sup>2</sup> ¼ 0, *k*<sup>3</sup> ¼ 1 and *k*<sup>4</sup> ¼ 2 corresponding to the relative *θm*j 4 *<sup>m</sup>*¼1. Then, the evolution of this interaction is depicted in **Figure 6**.

Similarly, for a 3–3 wave interaction one has:

$$A = \begin{pmatrix} \mathbf{1} & \mathbf{0} & -\mathbf{1} & -\mathbf{1} & \mathbf{0} & \mathbf{2} \\ \mathbf{0} & \mathbf{1} & \mathbf{2} & \mathbf{1} & \mathbf{0} & -\mathbf{1} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{1} \end{pmatrix},\tag{59}$$

and *k*<sup>1</sup> ¼ �1, *k*<sup>2</sup> ¼ �1*=*2, *k*<sup>3</sup> ¼ 0, *k*<sup>4</sup> ¼ 1*=*2, *k*<sup>5</sup> ¼ 1 and *k*<sup>6</sup> ¼ 3*=*2 to produce the following interaction pattern of **Figure 7**.

It is important to note here that the matrices *A* for the 2–2 and 3–3 solitons, are not arbitrarily chosen. The *N* � *N* maximal minors of *A* must be non-negative in order for the KP solution *q* to be regular for all *X*, *Y* and *T*. In the literature, such matrices are called totally non-negative matrices.

#### **3.2 Dark lump solitons**

So far, we have focused on structures obeying the effective KPII equation. These structures, namely the antidark stripe solitons, are quasi one-dimensional (1D) states, which are stable. The respective states that satisfy the effective KPI equation, namely dark stripe solitons, are expected to be transversely unstable – see, e.g., Ref. [36] for a stability analysis of KP line solitons.

Nevertheless, the KPI equation, that corresponds to the weakly nonlocal regime, supports stable, purely 2D, soliton states known as *lumps*. These solitons are weakly localized, namely they decay algebraically as ∣*x*∣, ∣*y*∣ ! ∞. The respective solitons of the nonlocal NLS, which are of the dark type, i.e., they have the form of dark lumps, can be effectively described as solutions of Eq. (45) for the field *U*:

$$U(\mathcal{X}, \mathcal{Y}, T) = 16 \frac{16q^2 \eta^2 - 4(\xi - 2k\eta)^2 + 1/q^2}{\left[16q^2 \eta^2 + 4(\xi - 2k\eta)^2 + 1/q^2\right]^2},\tag{60}$$

$$\xi = \mathcal{X} - 12\left(k^2 + q^2\right)T, \quad \eta = \mathcal{Y} - 12kT,$$

completely integrable model. The cKdV is a generic equation that can been used to describe cylindrical solitons in shallow water, plasmas, etc. [1]. As was shown in Ref. [37], the cKdV equation admits an exact soliton solution which, in the frame-

<sup>T</sup> sech<sup>2</sup> <sup>8</sup>*κ*<sup>3</sup>

where *η* and *ϱ*<sup>0</sup> are free parameters of the soliton. It is clear that Eq. (61) has the form of a sech2-pulse on top of a rational background. A clearer picture regarding the structure of the cKdV equation, can be obtained by means of an asymptotic asymptotic analysis [38, 39]. Indeed, as shown in these works, to leading-order, in the regime ∣T ∣≫∣*ϱ*∣, the primary wave *U*ð Þ *ϱ*, T , decays to zero at both upstream and downstream infinity. The primary wave has a form similar to that of Eq. (50), but with the following change of variables: X ↦ *ϱ* and *δ* ↦ *ϱ*0. However, there is a very important difference: *κ* now becomes a slowly-varying function of T , due to the presence of the term *U=*ð Þ 2T in the cKdV. Following the analysis of [38, 39], and

ffiffiffiffi T <sup>p</sup> <sup>þ</sup> *κϱ*

ffiffiffiffi T p þ *ϱ*<sup>0</sup>

� � (61)

, (62)

<sup>1</sup>*<sup>=</sup>*3, as follows

] nonlocal

work of Eq. (46), can be expressed as follows:

*Water Waves and Light: Two Unlikely Partners DOI: http://dx.doi.org/10.5772/intechopen.95431*

*<sup>U</sup>*ð Þ¼ *<sup>ϱ</sup>*, <sup>T</sup> *<sup>ϱ</sup>*

using the original coordinates, it can be concluded that:

where *κ*<sup>2</sup>

**Figure 9.**

**15**

amplitude and velocity varying as *t*

from Eqs. (50) and (62).

12T þ

2*κ*<sup>2</sup>

*<sup>κ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>κ</sup>*<sup>2</sup> 0 *t*0 *t* � �<sup>2</sup>*=*<sup>3</sup>

<sup>0</sup> is a constant setting the solitary wave amplitude at *t* ¼ *t*0. We can know express an approximate solution of Eqs. (2) and (3), for the polar chase, and for the primary solitary wave. This is of the form of Eq. (48), with the soliton

Notice that this approximate solution is a ring-shaped solitary wave, on top of

exhibiting either Kerr [40] or non-Kerr [41] nonlinearities, and were later observed in experiments [42]. On the other hand, ring anti-dark solitons were only predicted to occur in non-Kerr – e.g., saturable media [41, 43]. This picture is complemented

As in the Cartesian case, we have performed direct numerical simulations to examine the evolution of ring-shaped solitons in the nonlocal NLS model. First, in **Figure 9**, shown are 3D plots depicting the profiles of the ring dark solitons and ring antidark solitons. Here, we use *d* ¼ 1*=*5 for the dark soliton (*α*> 0) and *d* ¼ 1*=*3 for

cw background, which can be either dark (for *α*> 0) or anti-dark (for *α* <0). Notice, also that ring dark solitons were predicted to occur in optical media

by this analysis, according to which a relatively strong [i.e., *<sup>d</sup>*<sup>&</sup>gt; ð Þ <sup>1</sup>*=*4j*u*0<sup>j</sup> <sup>2</sup>

nonlinearity can also support ring anti-dark solitary waves.

*(Color online) typical ring dark (left) and anti-dark (right) solitons.*

�2*<sup>=</sup>*3, and the width varying as *t*

where *q* and *k* are free real parameters. Notice that solutions of this type have not yet been observed in water due to the fact that the surface tension is small.

Similarly to the case of antidark stripe solitons, once Eq. (60) is substituted into Eq. (48) gives rise to the approximate dark lump soliton of the nonlocal NLS. Using the relevant analytical expression, it is straightforward to prepare the corresponding initial condition by setting *t* ¼ 0, and numerically integrate Eqs. (2) and (3) to examine the evolution of dark lumps. In **Figure 8**, shown is the result of such a simulation for the dark lump; for this simulation, we have used the parameter values *u*<sup>0</sup> ¼ 1, *d* ¼ 1*=*5 (so that *α* ¼ 1*=*5>0, corresponding to dark soliton states), as well as *k* ¼ 0 and *q* ¼ 1. It can readily be observed that the dark lump solitons propagate undistorted up to *t* ¼ 20, which is the "numerical horizon" for this particular simulation. This result indicates that, similarly to antidark stripe solitons, dark lumps can also be supported by the nonlocal NLS system.

#### **3.3 Ring dark and antidark solitons**

We now turn our attention to the cKP Eq. (46), and focus on radially symmetric solutions that do not depend on Θ. In such a case, Eq. (46) reduces to a quasi-1D equation, namely the cylindrical Korteweg-de Vries (cKdV), which is also a

#### **Figure 8.**

*A typical dark lump solution, at t* ¼ 0*, with k* ¼ 0*, q* ¼ 1 *(top panel) and its evolution for t* ¼ 0*,* 10 *and* 20*. The initial condition shown in the top panel is given by Eq. (60).*

*Water Waves and Light: Two Unlikely Partners DOI: http://dx.doi.org/10.5772/intechopen.95431*

Nevertheless, the KPI equation, that corresponds to the weakly nonlocal regime, supports stable, purely 2D, soliton states known as *lumps*. These solitons are weakly localized, namely they decay algebraically as ∣*x*∣, ∣*y*∣ ! ∞. The respective solitons of the nonlocal NLS, which are of the dark type, i.e., they have the form of dark lumps,

16*q*2*η*<sup>2</sup> þ 4ð Þ *ξ* � 2*kη*

*<sup>ξ</sup>* <sup>¼</sup> <sup>X</sup> � <sup>12</sup> *<sup>k</sup>*<sup>2</sup> <sup>þ</sup> *<sup>q</sup>*<sup>2</sup> � �<sup>T</sup> , *<sup>η</sup>* <sup>¼</sup> <sup>Y</sup> � <sup>12</sup>*k*<sup>T</sup> ,

where *q* and *k* are free real parameters. Notice that solutions of this type have not yet been observed in water due to the fact that the surface tension is small.

corresponding initial condition by setting *t* ¼ 0, and numerically integrate Eqs. (2) and (3) to examine the evolution of dark lumps. In **Figure 8**, shown is the result of such a simulation for the dark lump; for this simulation, we have used the parameter values *u*<sup>0</sup> ¼ 1, *d* ¼ 1*=*5 (so that *α* ¼ 1*=*5>0, corresponding to dark soliton states), as well as *k* ¼ 0 and *q* ¼ 1. It can readily be observed that the dark lump solitons propagate undistorted up to *t* ¼ 20, which is the "numerical horizon" for this particular simulation. This result indicates that, similarly to antidark stripe solitons, dark lumps can also be supported by the nonlocal NLS system.

Similarly to the case of antidark stripe solitons, once Eq. (60) is substituted into Eq. (48) gives rise to the approximate dark lump soliton of the nonlocal NLS. Using

We now turn our attention to the cKP Eq. (46), and focus on radially symmetric solutions that do not depend on Θ. In such a case, Eq. (46) reduces to a quasi-1D equation, namely the cylindrical Korteweg-de Vries (cKdV), which is also a

*A typical dark lump solution, at t* ¼ 0*, with k* ¼ 0*, q* ¼ 1 *(top panel) and its evolution for t* ¼ 0*,* 10 *and* 20*.*

*The initial condition shown in the top panel is given by Eq. (60).*

h i<sup>2</sup> ,

<sup>2</sup> <sup>þ</sup> <sup>1</sup>*=q*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>1</sup>*=q*<sup>2</sup>

(60)

can be effectively described as solutions of Eq. (45) for the field *U*:

*Nonlinear Optics - From Solitons to Similaritons*

*<sup>U</sup>*ð Þ¼ <sup>X</sup>, <sup>Y</sup>, <sup>T</sup> <sup>16</sup> <sup>16</sup>*q*2*η*<sup>2</sup> � <sup>4</sup>ð Þ *<sup>ξ</sup>* � <sup>2</sup>*k<sup>η</sup>*

the relevant analytical expression, it is straightforward to prepare the

**3.3 Ring dark and antidark solitons**

**Figure 8.**

**14**

completely integrable model. The cKdV is a generic equation that can been used to describe cylindrical solitons in shallow water, plasmas, etc. [1]. As was shown in Ref. [37], the cKdV equation admits an exact soliton solution which, in the framework of Eq. (46), can be expressed as follows:

$$U(\varrho, T) = \frac{\varrho}{12T} + \frac{2\kappa^2}{T}\operatorname{sech}^2\left(\frac{8\kappa^3}{\sqrt{T}} + \frac{\kappa\varrho}{\sqrt{T}} + \varrho\_0\right) \tag{61}$$

where *η* and *ϱ*<sup>0</sup> are free parameters of the soliton. It is clear that Eq. (61) has the form of a sech2-pulse on top of a rational background. A clearer picture regarding the structure of the cKdV equation, can be obtained by means of an asymptotic asymptotic analysis [38, 39]. Indeed, as shown in these works, to leading-order, in the regime ∣T ∣≫∣*ϱ*∣, the primary wave *U*ð Þ *ϱ*, T , decays to zero at both upstream and downstream infinity. The primary wave has a form similar to that of Eq. (50), but with the following change of variables: X ↦ *ϱ* and *δ* ↦ *ϱ*0. However, there is a very important difference: *κ* now becomes a slowly-varying function of T , due to the presence of the term *U=*ð Þ 2T in the cKdV. Following the analysis of [38, 39], and using the original coordinates, it can be concluded that:

$$
\kappa^2 = \kappa\_0^2 \left(\frac{t\_0}{t}\right)^{2/3},
\tag{62}
$$

where *κ*<sup>2</sup> <sup>0</sup> is a constant setting the solitary wave amplitude at *t* ¼ *t*0. We can know express an approximate solution of Eqs. (2) and (3), for the polar chase, and for the primary solitary wave. This is of the form of Eq. (48), with the soliton amplitude and velocity varying as *t* �2*<sup>=</sup>*3, and the width varying as *t* <sup>1</sup>*<sup>=</sup>*3, as follows from Eqs. (50) and (62).

Notice that this approximate solution is a ring-shaped solitary wave, on top of cw background, which can be either dark (for *α*> 0) or anti-dark (for *α* <0). Notice, also that ring dark solitons were predicted to occur in optical media exhibiting either Kerr [40] or non-Kerr [41] nonlinearities, and were later observed in experiments [42]. On the other hand, ring anti-dark solitons were only predicted to occur in non-Kerr – e.g., saturable media [41, 43]. This picture is complemented by this analysis, according to which a relatively strong [i.e., *<sup>d</sup>*<sup>&</sup>gt; ð Þ <sup>1</sup>*=*4j*u*0<sup>j</sup> <sup>2</sup> ] nonlocal nonlinearity can also support ring anti-dark solitary waves.

As in the Cartesian case, we have performed direct numerical simulations to examine the evolution of ring-shaped solitons in the nonlocal NLS model. First, in **Figure 9**, shown are 3D plots depicting the profiles of the ring dark solitons and ring antidark solitons. Here, we use *d* ¼ 1*=*5 for the dark soliton (*α*> 0) and *d* ¼ 1*=*3 for

**Figure 9.** *(Color online) typical ring dark (left) and anti-dark (right) solitons.*

well as ring dark and antidark solitons. We found that all these states do exist in the framework of the nonlocal NLS, and can propagate undistorted up to times set by

At this point, it is also relevant to comment on the possibility of observing the predicted patterns in real experiments. In that regard, first we consider the case of antidark solitons. The observation of these states took place some time ago, as it was first reported in Ref. [44]; this fact indicates that the necessary experimental setting is already available. As concerns the experimental set up needed for the observation of the Y-, X- and H-waves and other related patterns, it may be similar to that of Ref. [44]. In particular, one may employ at first a cw laser beam, which is split into two parts via a beam-splitter. One branch goes through a cavity system to form a pulse (as happens in typical pulsed lasers); this pulse branch undergoes phaseengineering, i.e., passes through a phase mask so that the characteristic phase jump of the antidark soliton is inscribed. Then, the cw and the phase-engineered pulse are incoherently coupled inside the nonlocal medium, e.g., a nematic liquid crystal. This process forms one antidark soliton, as in Ref. [44]. To observe the Y-, X- or Hpatterns predicted above, two such antidark solitons have to be combined inside the crystal. The angle between the two incident beams, which should be appropriately chosen so that a specific pattern be formed, can be controlled by a rotating mirror in

As concerns the possibility of the experimental observation of the other soliton states that were predicted by our analysis, we note the following. First, regarding the weakly localized dark lump solitons, they can be experimentally observed as follows. In a real experiment, it is straightforward to create a dark soliton by using, e.g., a proper phase mask (see, e.g., the review [45] and references therein); this can also be done even in the case of *ring dark solitons* [46]. Such structures are prone to the transverse modulational instability in 2D, and are known to decay into vortices [45]. However, as was predicted in Ref. [19], sufficiently weak gray solitons can decay to *dark lumps* satisfying an effective KPI equation. This result suggests that, to observe dark lumps, one only need to embed a shallow dark soliton (like the ones we considered herein) into a 2D space, e.g., in a planar waveguide; then, the transverse instability will set in, and dark lump solitons will emerge. Finally, regarding the ring solitons, as mentioned above, they have already been observed in a variety of experiments (see, e.g., Ref. [46] and the review [45]) and, thus they can readily be created, e.g., in thermal media or in nematic liquid crystals. Thus, a variety of patterns that can be observed in water may also emerge in optics.

The obtained results relied on the formal reduction of the nonlocal NLS model to

the KP (cKP) equation. It would be interesting to investigate if other types of nonlocality, arising, e.g., in the context of chromium Bose-Einstein condensates [47], or in defocusing colloidal media [48] can support relevant soliton states. Such

studies are in progress and pertinent results will be presented elsewhere.

the simulations.

**17**

one of the branches of the beam-splitter.

*Water Waves and Light: Two Unlikely Partners DOI: http://dx.doi.org/10.5772/intechopen.95431*

#### **Figure 10.**

*(Color online) the evolution of the dark (top) and anti-dark (bottom) ring solitons of Figure 9.*

the antidark (*α*<0), while the rest of the parameters are set equal to unity except *ε* ¼ 0*:*2 as before. Furthermore, both solitons have an initial radius of *r*ð Þ¼ 0 10.

In addition, in **Figure 10**, we show the evolution of the ring solitons depicted in **Figure 9**. Once again, it is observed, that both types of ring solitons propagate undistorted up to end of the simulation (*t* ¼ 10), which indicates that these states are also supported by the nonlocal NLS.

## **4. Conclusions and discussion**

In conclusion, we have studied a 2D defocusing NLS model with a spatially nonlocal nonlinearity. The considered model is relevant to optical beam propagation in a variety of physical settings, including thermal media, plasmas, and nematic liquid crystals. We have analytically treated this nonlocal NLS by means of a twostage multiscale analysis. This led to a Boussinesq/Benney-Luke-type equation, and a KP equation (in both Cartesian and polar geometries), which are models arising in the context of shallow water waves. This analysis revealed that the nonlocal NLS is characterized by an *effective surface tension*, which is related to the degree of nonlocality: small (large) effective surface tension corresponds to the strong (weak) nonlocality regime, whereby the asymptotic reduction leads to a KPII (KPI) equation.

Apart from the type of KP (KPII or KPI), the effective surface tension also controls the type of soliton: antidark (dark) solitons on top (off) the cw pedestal can only be formed in the strong (weak) nonlocality regime. We have thus found a number of different soliton states that can be supported by the nonlocal NLS. These include antidark stripe solitons, dark lump solitons, as well as ring dark and antidark solitons (in the polar case). We also used numerical simulations to examine the existence, evolution, as well as interaction dynamics. For the latter, we focused on the case of antidark solitons, which were shown to form a plethora of patterns coming out of their collisions. These include Y-, X-, H-shaped waveforms, as well as more complicated patterns involving more than two solitons; all these, have been observed in shallow water [5], a fact that highlights the deeper connection between water waves and light! In addition, we examined, by means of direct simulations, the existence and propagation of other soliton states, namely dark lump solitons, as *Water Waves and Light: Two Unlikely Partners DOI: http://dx.doi.org/10.5772/intechopen.95431*

well as ring dark and antidark solitons. We found that all these states do exist in the framework of the nonlocal NLS, and can propagate undistorted up to times set by the simulations.

At this point, it is also relevant to comment on the possibility of observing the predicted patterns in real experiments. In that regard, first we consider the case of antidark solitons. The observation of these states took place some time ago, as it was first reported in Ref. [44]; this fact indicates that the necessary experimental setting is already available. As concerns the experimental set up needed for the observation of the Y-, X- and H-waves and other related patterns, it may be similar to that of Ref. [44]. In particular, one may employ at first a cw laser beam, which is split into two parts via a beam-splitter. One branch goes through a cavity system to form a pulse (as happens in typical pulsed lasers); this pulse branch undergoes phaseengineering, i.e., passes through a phase mask so that the characteristic phase jump of the antidark soliton is inscribed. Then, the cw and the phase-engineered pulse are incoherently coupled inside the nonlocal medium, e.g., a nematic liquid crystal. This process forms one antidark soliton, as in Ref. [44]. To observe the Y-, X- or Hpatterns predicted above, two such antidark solitons have to be combined inside the crystal. The angle between the two incident beams, which should be appropriately chosen so that a specific pattern be formed, can be controlled by a rotating mirror in one of the branches of the beam-splitter.

As concerns the possibility of the experimental observation of the other soliton states that were predicted by our analysis, we note the following. First, regarding the weakly localized dark lump solitons, they can be experimentally observed as follows. In a real experiment, it is straightforward to create a dark soliton by using, e.g., a proper phase mask (see, e.g., the review [45] and references therein); this can also be done even in the case of *ring dark solitons* [46]. Such structures are prone to the transverse modulational instability in 2D, and are known to decay into vortices [45]. However, as was predicted in Ref. [19], sufficiently weak gray solitons can decay to *dark lumps* satisfying an effective KPI equation. This result suggests that, to observe dark lumps, one only need to embed a shallow dark soliton (like the ones we considered herein) into a 2D space, e.g., in a planar waveguide; then, the transverse instability will set in, and dark lump solitons will emerge. Finally, regarding the ring solitons, as mentioned above, they have already been observed in a variety of experiments (see, e.g., Ref. [46] and the review [45]) and, thus they can readily be created, e.g., in thermal media or in nematic liquid crystals. Thus, a variety of patterns that can be observed in water may also emerge in optics.

The obtained results relied on the formal reduction of the nonlocal NLS model to the KP (cKP) equation. It would be interesting to investigate if other types of nonlocality, arising, e.g., in the context of chromium Bose-Einstein condensates [47], or in defocusing colloidal media [48] can support relevant soliton states. Such studies are in progress and pertinent results will be presented elsewhere.

the antidark (*α*<0), while the rest of the parameters are set equal to unity except *ε* ¼ 0*:*2 as before. Furthermore, both solitons have an initial radius of *r*ð Þ¼ 0 10. In addition, in **Figure 10**, we show the evolution of the ring solitons depicted in

In conclusion, we have studied a 2D defocusing NLS model with a spatially nonlocal nonlinearity. The considered model is relevant to optical beam propagation in a variety of physical settings, including thermal media, plasmas, and nematic liquid crystals. We have analytically treated this nonlocal NLS by means of a twostage multiscale analysis. This led to a Boussinesq/Benney-Luke-type equation, and a KP equation (in both Cartesian and polar geometries), which are models arising in the context of shallow water waves. This analysis revealed that the nonlocal NLS is characterized by an *effective surface tension*, which is related to the degree of nonlocality: small (large) effective surface tension corresponds to the strong (weak) nonlocality regime, whereby the asymptotic reduction leads to a KPII (KPI)

Apart from the type of KP (KPII or KPI), the effective surface tension also controls the type of soliton: antidark (dark) solitons on top (off) the cw pedestal can only be formed in the strong (weak) nonlocality regime. We have thus found a number of different soliton states that can be supported by the nonlocal NLS. These

antidark solitons (in the polar case). We also used numerical simulations to examine the existence, evolution, as well as interaction dynamics. For the latter, we focused on the case of antidark solitons, which were shown to form a plethora of patterns coming out of their collisions. These include Y-, X-, H-shaped waveforms, as well as more complicated patterns involving more than two solitons; all these, have been observed in shallow water [5], a fact that highlights the deeper connection between water waves and light! In addition, we examined, by means of direct simulations, the existence and propagation of other soliton states, namely dark lump solitons, as

include antidark stripe solitons, dark lump solitons, as well as ring dark and

**Figure 9**. Once again, it is observed, that both types of ring solitons propagate undistorted up to end of the simulation (*t* ¼ 10), which indicates that these states

*(Color online) the evolution of the dark (top) and anti-dark (bottom) ring solitons of Figure 9.*

are also supported by the nonlocal NLS.

*Nonlinear Optics - From Solitons to Similaritons*

**4. Conclusions and discussion**

equation.

**16**

**Figure 10.**

*Nonlinear Optics - From Solitons to Similaritons*

## **Author details**

Georgios N. Koutsokostas<sup>1</sup> \*, Theodoros P. Horikis<sup>2</sup> , Dimitrios J. Frantzeskakis<sup>1</sup> , Nalan Antar<sup>3</sup> and İlkay Bakırtaş 3

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Cambridge University Press;

1 Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece

2 Department of Mathematics, University of Ioannina, Ioannina 45110, Greece

3 Department of Mathematics, Istanbul Technical University, Maslak 34469, Istanbul, Turkey

\*Address all correspondence to: koutsokwstasgewrgios@gmail.com

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Water Waves and Light: Two Unlikely Partners DOI: http://dx.doi.org/10.5772/intechopen.95431*

## **References**

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**Author details**

Georgios N. Koutsokostas<sup>1</sup>

Athens 15784, Greece

Istanbul, Turkey

**18**

Nalan Antar<sup>3</sup> and İlkay Bakırtaş

*Nonlinear Optics - From Solitons to Similaritons*

provided the original work is properly cited.

\*, Theodoros P. Horikis<sup>2</sup>

1 Department of Physics, University of Athens, Panepistimiopolis, Zografos,

2 Department of Mathematics, University of Ioannina, Ioannina 45110, Greece

3 Department of Mathematics, Istanbul Technical University, Maslak 34469,

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: koutsokwstasgewrgios@gmail.com

3

, Dimitrios J. Frantzeskakis<sup>1</sup>

,

[22] Johnson RS. Water waves and Korteweg-de Vries equations. J Fluid Mech. 1980;97:701–719.

[23] Johnson RS. A modern introduction to the mathematical theory of water waves. Cambridge University Press; 1997.

[24] Benney DJ, Luke JC. On the interactions of permanent waves of finite amplitude. J Math and Phys. 1964; 43:309–313.

[25] Rotschild C, Cohen O, Manela O, Segev M, Carmon T. Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortexring solitons. Phys Rev Lett. 2005;95: 213904.

[26] Krolikowski W, Bang O, Nikolov NI, Neshev D, Wyller J, Rasmussen JJ, et al. Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media. J Opt B: Quantum Semiclass Opt. 2004;6:S288–S294.

[27] Ghofraniha N, Conti C, Ruocco G, Trillo S. Shocks in nonlocal media. Phys Rev Lett. 2007;99:043903.

[28] Conti C, Fratalocchi A, Peccianti M, Ruocco G, Trillo S. Observation of a gradient catastrophe generating solitons. Phys Rev Lett. 2009;102: 083902.

[29] Litvak AG, Mironov VA, Fraiman GM, Yunakovskii AD. Thermal self-effect of wave beams in a plasma with a nonlocal nonlinearity. Sov J Plasma Phys. 1975;1:60–71.

[30] Yakimenko AI, Zaliznyak YA, Kivshar YS. Stable vortex solitons in nonlocal self-focusing nonlinear media. Phys Rev E. 2005;71:065603(R).

[31] Conti C, Peccianti M, Assanto G. Route to nonlocality and observation of accessible solitons. Phys Rev Lett. 2003; 91:073901.

collisions of ring dark solitons. Phys

*Water Waves and Light: Two Unlikely Partners DOI: http://dx.doi.org/10.5772/intechopen.95431*

[44] Coskun TH, Christodoulides DN, Kim YR, Chen Z, Soljacic M, M S. Bright Spatial Solitons on a Partially Incoherent Background. Phys Rev Lett. 2000;84:

[45] Kivshar YS, Luther-Davies B. Dark

applications. Phys Rep. 1998;298:81–197.

[46] Dreischuh A, Neshev D, Paulus GG, Grasbon F, Walther H. Ring dark solitary waves: Experiment versus theory. Phys Rev E. 2002;66:066611.

[47] Lahaye T, Menotti C, Santos L, Lewenstein M, Pfau T. The physics of dipolar bosonic quantum gases. Rep

[48] An X, Marchant TR, Smyth NF. Optical dispersive shock waves in defocusing colloidal media. Physica D.

Prog Phys. 2009;72:126401.

2017;342:45–56.

**21**

Lett A. 2001;285:157–164.

optical solitons: physics and

2374–2377.

[32] Assanto G. Nematicons: Spatial Optical Solitons in Nematic Liquid Crystals. New Jersey: Wiley-Blackwell; 2012.

[33] Ablowitz MJ, Clarkson PA. Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press; Cambridge, 1991.

[34] Horikis TP. Small-amplitude defocusing nematicons. J Phys A: Math Theor. 2015;48:02FT01.

[35] Horikis TP, Frantzeskakis DJ. Ring dark and antidark solitons in nonlocal media. Opt Lett. 2016;41:583–586.

[36] Ablowitz MJ, Segur H. Solitons and the inverse scattering transform. SIAM; 1981.

[37] Hirota R. Exact solutions to the equation describing "cylindrical solitons". Phys Lett A. 1979;71:393–394.

[38] Johnson RS. A note on an asymptotic solution of the cylindrical Korteweg-de Vries equation. Wave Motion. 1999;30:1–16.

[39] Ko K, Kuehl HH. Cylindrical and spherical Korteweg-deVries solitary waves. Phys Fluids. 1979;22:1343–1348.

[40] Kivshar YS, Yang X. Ring dark solitons. Phys Rev E. 1994;50:R40–R43.

[41] Frantzeskakis DJ, Malomed BA. Multiscale expansions for a generalized cylindrical nonlinear Schrödinger equation. Phys Lett A. 1999;264: 179–185.

[42] Dreischuh A, Neshev D, Paulus GG, Grasbon F, Walther H. Ring dark solitary waves: Experiment versus theory. Phys Rev E. 2002;66:066611.

[43] Nistazakis HE, Frantzeskakis DJ, Malomed BA, Kevrekidis PG. Head-on *Water Waves and Light: Two Unlikely Partners DOI: http://dx.doi.org/10.5772/intechopen.95431*

collisions of ring dark solitons. Phys Lett A. 2001;285:157–164.

[22] Johnson RS. Water waves and Korteweg-de Vries equations. J Fluid

[24] Benney DJ, Luke JC. On the interactions of permanent waves of finite amplitude. J Math and Phys. 1964;

[25] Rotschild C, Cohen O, Manela O, Segev M, Carmon T. Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortexring solitons. Phys Rev Lett. 2005;95:

[26] Krolikowski W, Bang O, Nikolov NI, Neshev D, Wyller J, Rasmussen JJ, et al. Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media. J Opt B: Quantum Semiclass Opt. 2004;6:S288–S294.

Rev Lett. 2007;99:043903.

[29] Litvak AG, Mironov VA,

Plasma Phys. 1975;1:60–71.

[30] Yakimenko AI, Zaliznyak YA, Kivshar YS. Stable vortex solitons in nonlocal self-focusing nonlinear media. Phys Rev E. 2005;71:065603(R).

[31] Conti C, Peccianti M, Assanto G. Route to nonlocality and observation of

[27] Ghofraniha N, Conti C, Ruocco G, Trillo S. Shocks in nonlocal media. Phys

[28] Conti C, Fratalocchi A, Peccianti M, Ruocco G, Trillo S. Observation of a gradient catastrophe generating solitons. Phys Rev Lett. 2009;102:

Fraiman GM, Yunakovskii AD. Thermal self-effect of wave beams in a plasma with a nonlocal nonlinearity. Sov J

[23] Johnson RS. A modern introduction to the mathematical theory of water waves. Cambridge University Press;

*Nonlinear Optics - From Solitons to Similaritons*

accessible solitons. Phys Rev Lett. 2003;

[33] Ablowitz MJ, Clarkson PA. Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press; Cambridge, 1991.

[34] Horikis TP. Small-amplitude defocusing nematicons. J Phys A: Math

[35] Horikis TP, Frantzeskakis DJ. Ring dark and antidark solitons in nonlocal media. Opt Lett. 2016;41:583–586.

[36] Ablowitz MJ, Segur H. Solitons and the inverse scattering transform. SIAM;

[37] Hirota R. Exact solutions to the equation describing "cylindrical

[38] Johnson RS. A note on an

Motion. 1999;30:1–16.

solitons". Phys Lett A. 1979;71:393–394.

asymptotic solution of the cylindrical Korteweg-de Vries equation. Wave

[39] Ko K, Kuehl HH. Cylindrical and spherical Korteweg-deVries solitary waves. Phys Fluids. 1979;22:1343–1348.

[40] Kivshar YS, Yang X. Ring dark solitons. Phys Rev E. 1994;50:R40–R43.

[41] Frantzeskakis DJ, Malomed BA. Multiscale expansions for a generalized cylindrical nonlinear Schrödinger equation. Phys Lett A. 1999;264:

[42] Dreischuh A, Neshev D, Paulus GG, Grasbon F, Walther H. Ring dark solitary waves: Experiment versus theory. Phys Rev E. 2002;66:066611.

[43] Nistazakis HE, Frantzeskakis DJ, Malomed BA, Kevrekidis PG. Head-on

Theor. 2015;48:02FT01.

[32] Assanto G. Nematicons: Spatial Optical Solitons in Nematic Liquid Crystals. New Jersey: Wiley-Blackwell;

91:073901.

2012.

1981.

179–185.

Mech. 1980;97:701–719.

1997.

43:309–313.

213904.

083902.

**20**

[44] Coskun TH, Christodoulides DN, Kim YR, Chen Z, Soljacic M, M S. Bright Spatial Solitons on a Partially Incoherent Background. Phys Rev Lett. 2000;84: 2374–2377.

[45] Kivshar YS, Luther-Davies B. Dark optical solitons: physics and applications. Phys Rep. 1998;298:81–197.

[46] Dreischuh A, Neshev D, Paulus GG, Grasbon F, Walther H. Ring dark solitary waves: Experiment versus theory. Phys Rev E. 2002;66:066611.

[47] Lahaye T, Menotti C, Santos L, Lewenstein M, Pfau T. The physics of dipolar bosonic quantum gases. Rep Prog Phys. 2009;72:126401.

[48] An X, Marchant TR, Smyth NF. Optical dispersive shock waves in defocusing colloidal media. Physica D. 2017;342:45–56.

**Chapter 2**

**Abstract**

**1. Introduction**

DNLS equation.

**23**

*Zhou Guo-Quan*

Soliton and Rogue-Wave Solutions

of Derivative Nonlinear

Schrödinger Equation - Part 1

Based upon different methods such as a newly revised version of inverse scattering transform, Marchenko formalism, and Hirota's bilinear derivative transform, this chapter aims to study and solve the derivative nonlinear Schrödinger (DNLS for brevity) equation under vanishing boundary condition (VBC for brevity). The explicit one-soliton and multi-soliton solutions had been derived by some algebra techniques for the VBC case. Meanwhile, the asymptotic behaviors of

**Keywords:** soliton, nonlinear equation, derivative nonlinear Schrödinger equation, inverse scattering transform, Zakharov-Shabat equation, Marchenko formalism,

Derivative nonlinear Schrödinger (DNLS for brevity) equation is one of the several rare kinds of integrable nonlinear models. Research of DNLS equation has not only mathematic interest and significance, but also important physical application background. It was first found that the Alfven waves in space plasma [1–3] can be modeled with DNLS equation. The modified nonlinear Schrödinger (MNLS for brevity) equation, which is used to describe the sub-picosecond pulses in single mode optical fibers [4–6], is actually a transformed version of DNLS equation. The weak nonlinear electromagnetic waves in ferromagnetic, anti-ferromagnetic, or dielectric systems [5–9] under external magnetic fields can also be modeled by

Although DNLS equation is similar to NLS equation in form, it does not belong

to the famous AKNS hierarchy at all. As is well known, a nonlinear integrable equation can be transformed to a pair of Lax equation satisfied by its Jost functions, the original nonlinear equation is only the compatibility condition of the Lax pair, that is, the so-called zero-curvature condition. Another fact had been found by some scholars that those nonlinear integrable equations which have the same first operator of the Lax pair belong to the same hierarchy and can deal with the same inverse scattering transform (IST for brevity). As a matter of fact, the DNLS

those multi-soliton solutions had been analyzed and discussed in detail.

Hirota's bilinear derivative transform, rogue wave

## **Chapter 2**
