Bright, Dark, and Kink Solitary Waves in a Cubic-Quintic-Septic-Nonical Medium

*Mati Youssoufa, Ousmanou Dafounansou and Alidou Mohamadou*

## **Abstract**

In this chapter, evolution of light beams in a cubic-quintic-septic-nonical medium is investigated. As the model equation, an extended form of the wellknown nonlinear Schrödinger (NLS) equation is taken into account. By the use of a special ansatz, exact analytical solutions describing bright/dark and kink solitons are constructed. The existence of the wave solutions is discussed in a parameter regime. Moreover, the stability properties of the obtained solutions are investigated, and by employing Stuart and DiPrima's stability analysis method, an analytical expression for the modulational stability is found.

**Keywords:** higher-order nonlinear Schrödinger equation, spatial solitons, stability analysis method, modulational instability, optical fibers

## **1. Introduction**

The study of spatial solitons in the field of fiber-optical communication has attracted considerable interest in recent years. In a uniform nonlinear fiber, soliton can propagate over relatively long distance without any considerable attenuation. The formation of optical solitons in optical fibers results from an exact balancing between the diffraction and/or group velocity dispersion (GVD) and the self-phase modulation (SPM). The theorical prediction of a train of soliton pulses from a continuouswave (CW) light in optical fibers was first suggested by Hasegawa and Tappert [1, 2] and first experimentally demonstrated by Mollenauer et al. [3] in single-mode fibers in the case of negative GVD, in liquid *CS*<sup>2</sup> by Barthelemy et al. in 1985 [4]. In nonlinear optic, optical solitons are localized electromagnetic waves that transmit in nonlinear Kerr or non-Kerr media with dispersion or (and) diffraction without any change in shapes. In nonlinear media, the dynamics of spatial optical solitons is governed by the well-known nonlinear Schrödinger (NLS) equation. Depending on the signs of GVD, the NLS equation admits two distinct types of soliton, namely, bright and dark solitons. The bright soliton exists in the regime of anomalous GVD, and the dark soliton arises in the regime of normal GVD. The physics governing the soliton differs depending on whether one considers a bright or a dark soliton and

accordingly features distinct applications [5–8]. The unique property of optical solitons, either bright or dark, is their particle-like behavior in interaction [9].

Recently, the study of modulational instability (MI) in non-Kerr media has receiving particular attention. MI is a fundamental and ubiquitous process that appears in most nonlinear systems [6, 9, 34–37]. This instability is referred to as modulation instability because it leads to a spontaneous temporal modulation of the CW beam and transforms it into a pulse train. During this process, small perturbations upon a uniform intensity beam grow exponentially due to the interplay between nonlinearity and dispersion or diffraction. As a result, under specific conditions, a CW light often breaks up into trains of ultrashort solitons like pulses [9]. To date, there has not been any report of MI in the cubic-quintic-septic-

*Bright, Dark, and Kink Solitary Waves in a Cubic-Quintic-Septic-Nonical Medium*

Our study will be focused on the analysis of solitary wave's solutions of systems described by the higher-order NLSE named CQSNNLSE. We will discuss the model with higher-order nonlinearities and explore the dynamics of bright, dark, and kink soliton solutions. Finally, the linear stability analysis of the MI is formulated, and the analytical expression of the gain of MI is obtained. Moreover, the typical

The dynamics of (1 + 1)-dimensional (one spatial and one temporal variables) spatial optical solitons is the well-known nonlinear Schrödinger equation. If we consider the higher-order effects, an extended model is required, and the propagation of optical pulses through the highly nonlinear waveguides can be described by

nonical-nonlinear Schrödinger equation (CQSNNLSE).

*DOI: http://dx.doi.org/10.5772/intechopen.92819*

*Ez* ¼ *iα***1***Ett* þ *iα***2**|*E*|

For example, Eq. (1) with *<sup>α</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup>

employ the following transformation:

*θζζ* <sup>¼</sup> *<sup>k</sup>* <sup>þ</sup> *<sup>α</sup>***1***ω***<sup>2</sup>** *α***1**

**2. Model equation**

the CQSNNLSE:

nonical media.

parts, one obtains

**213**

outcomes of the nonlinear development of the MI are reported.

**2**

2

ically the stability conditions of one-dimensional spatial solitons [38]. Recently, Eq. (1) with *α*<sup>5</sup> ¼ 0 was analyzed for systems that are valid for several types of septic nonlinear materials [28]. Here, we consider arbitrary parameters

*α <sup>j</sup>*ð Þ *j* ¼ 1, 2, 3, 4, 5 for the sake of a general analysis that is valid for several types of

To obtain the exact analytic optical solitary-wave solutions of Eq. (1), we can

Here, *θ ζ*ð Þ is a real function and *β* is a real constant to be determined. Upon substituting Eq. (2) into Eq. (1) and separating the real and imaginary

> *<sup>θ</sup>* � *<sup>α</sup>***<sup>2</sup>** *α***1**

*<sup>θ</sup>***<sup>3</sup>** � *<sup>α</sup>***<sup>3</sup>** *α***1**

*E* þ *iα***3**|*E*|

where *E z*ð Þ , *t* is the slowly varying envelope of the electric field, the subscripts *z* and *t* are the spatial and temporal partial derivatives in the frame moving with the pulsed solutions, *α*<sup>1</sup> is the parameter of diffraction or dispersion, and *α*2, *α*3, *α*4, and *α*<sup>5</sup> are the cubic, quintic, septic, and nonical nonlinear terms, respectively. This model is relevant to some applications in which higher-order nonlinearities are important.

**<sup>4</sup>***<sup>E</sup>* <sup>þ</sup> *<sup>i</sup>α***4**|*E*<sup>|</sup>

*E z* ð Þ¼ ,*<sup>t</sup> <sup>θ</sup>*ð Þ *<sup>t</sup>* <sup>þ</sup> *<sup>β</sup><sup>z</sup> ei kz* ð Þ �*ω<sup>t</sup>* <sup>¼</sup> *θ ζ*ð Þ*ei kz* ð Þ �*ω<sup>t</sup> :* (2)

*<sup>θ</sup>***<sup>5</sup>** � *<sup>α</sup>***<sup>4</sup>** *α***1**

**6**

*E* þ *iα***5**|*E*|

, *α*<sup>4</sup> ¼ 1, and *α*<sup>5</sup> ¼ 0 was used to study numer-

*β* ¼ **2***α***1***ω*, (3)

*<sup>θ</sup>***<sup>7</sup>** � *<sup>α</sup>***<sup>5</sup>** *α***1**

*θ***<sup>9</sup>**, (4)

**8**

*E*, (1)

In addition to fundamental bright and dark solitons, various other forms and shapes of solitary waves can appear in nonlinear media. Kink solitons, for example, are an important class of solitons which may propagate in nonlinear media exhibiting higher-order effects such as third-order dispersion, self-steepening, higher-order nonlinearity, and intrapulse stimulated Raman scattering. In the setting of nonlinear optics, a kink soliton represents a shock front that propagates undistorted inside the dispersive nonlinear medium [10]. This type of solitons has been studied extensively, both analytically and numerically [11–13]. These spatial soliton solutions can maintain their overall shapes but allow their widths and amplitudes and the pulse center to change according to the management of the system's parameters, such as the dispersion, nonlinearity, gain, and so on [14].

The cubic nonlinear Schrödinger equation (CNLSE) has been widely used to model the propagation of light pulse in material's systems involving third-order susceptibility *χ*ð Þ<sup>3</sup> , though, for moderate pulse intensity, the higher-order nonlinearities are related to higher-order nonlinear susceptibilities (nonlinear responses) of a material. For example, the cubic-quintic-nonlinear Schrödinger equation (CQNLSE) models materials with fifth-order susceptibility *χ*ð Þ<sup>5</sup> . This kind of nonlinearity (cubicquintic CQ) is named as *parabolic law nonlinearity* and existing in nonlinear media such as the p-toluene sulfonate (PTS) crystals. The parabolic law can closely describe the nonlinear interaction between the high-frequency Langmuir waves and the ion acoustic waves by ponderomotive forces [15, 16], in a region of reduced plasma density, and the nonlinear interaction between Langmuir waves and electrons. In addition, CQ was experimentally proposed as an empirical description of special semiconductor (e.g., AlGaAs, CdS, etc.) waveguides and semiconductor-doped glasses, particularly for the *CdSxSe*<sup>1</sup>�*<sup>x</sup>*-doped glass, which exhibit a significant fifthorder susceptibilities *χ*ð Þ<sup>5</sup> as experimentally reported earlier [17, 18]. Moreover, using high laser intensity, the saturation of nonlinearity has been established experimentally in many materials such as nonlinear organic polymers, semiconductor-doped glasses, and so on, which have the property that their absorption coefficient decreases [19]. More generally, a self-defocusing *χ*ð Þ<sup>5</sup> usually accounts for the saturation of *χ*ð Þ<sup>3</sup> .

In recent years, many influential works have devoted to construct exact analytical solutions of CQNLSE, such as the pioneering work of Serkin et al. [20]. In particular, Dai et al. [21–25] obtained exact self-similar solutions (similaritons), their nonlinear tunneling effects of the generalized CQNLSE, and their higherdimensional forms with spatially inhomogeneous group velocity dispersion, cubicquintic nonlinearity, and amplification or attenuation.

Since the measurement of third-, fifth-, and seventh-order nonlinearities of silver nanoplatelet colloids using a femtosecond laser [26], an extension of nonlinear Schrödinger equation including the cubic-quintic-septic nonlinearity was used to model the propagation of spatial solitons. In [27], for example, the authors performed numerical calculations based on higher-order nonlinearity parameters including seventh-order susceptibility *χ*ð Þ<sup>7</sup> (a chalcogenide glass is an example). This seeds several motivations to discover new features of solitons with combined effects of higher-order nonlinear parameters. In this regard, Houria et al. [28] constructed dark spatial solitary waves in a cubic-quintic-septic-nonlinear medium, with a profile in a functional form given in terms of "*sech*<sup>2</sup>*=*3 ". They have also investigated chirped solitary pulses for a derivative nonical-NLS equation on a CW background [29]. It is obvious to notice that the contributions of the higher-order nonlinearities can give way to generate stable solitons in homogeneous isotropic media and influence many aspects of filamentation in gases and condensed matters [30–33].

*Bright, Dark, and Kink Solitary Waves in a Cubic-Quintic-Septic-Nonical Medium DOI: http://dx.doi.org/10.5772/intechopen.92819*

Recently, the study of modulational instability (MI) in non-Kerr media has receiving particular attention. MI is a fundamental and ubiquitous process that appears in most nonlinear systems [6, 9, 34–37]. This instability is referred to as modulation instability because it leads to a spontaneous temporal modulation of the CW beam and transforms it into a pulse train. During this process, small perturbations upon a uniform intensity beam grow exponentially due to the interplay between nonlinearity and dispersion or diffraction. As a result, under specific conditions, a CW light often breaks up into trains of ultrashort solitons like pulses [9]. To date, there has not been any report of MI in the cubic-quintic-septicnonical-nonlinear Schrödinger equation (CQSNNLSE).

Our study will be focused on the analysis of solitary wave's solutions of systems described by the higher-order NLSE named CQSNNLSE. We will discuss the model with higher-order nonlinearities and explore the dynamics of bright, dark, and kink soliton solutions. Finally, the linear stability analysis of the MI is formulated, and the analytical expression of the gain of MI is obtained. Moreover, the typical outcomes of the nonlinear development of the MI are reported.

### **2. Model equation**

accordingly features distinct applications [5–8]. The unique property of optical soli-

In addition to fundamental bright and dark solitons, various other forms and shapes of solitary waves can appear in nonlinear media. Kink solitons, for example,

tons, either bright or dark, is their particle-like behavior in interaction [9].

*Nonlinear Optics - From Solitons to Similaritons*

are an important class of solitons which may propagate in nonlinear media exhibiting higher-order effects such as third-order dispersion, self-steepening, higher-order nonlinearity, and intrapulse stimulated Raman scattering. In the setting of nonlinear optics, a kink soliton represents a shock front that propagates undistorted inside the dispersive nonlinear medium [10]. This type of solitons has been studied extensively, both analytically and numerically [11–13]. These spatial soliton solutions can maintain their overall shapes but allow their widths and amplitudes and the pulse center to change according to the management of the system's parameters, such as the dispersion, nonlinearity, gain, and so on [14]. The cubic nonlinear Schrödinger equation (CNLSE) has been widely used to model the propagation of light pulse in material's systems involving third-order susceptibility *χ*ð Þ<sup>3</sup> , though, for moderate pulse intensity, the higher-order nonlinearities are related to higher-order nonlinear susceptibilities (nonlinear responses) of a material. For example, the cubic-quintic-nonlinear Schrödinger equation (CQNLSE) models materials with fifth-order susceptibility *χ*ð Þ<sup>5</sup> . This kind of nonlinearity (cubicquintic CQ) is named as *parabolic law nonlinearity* and existing in nonlinear media such as the p-toluene sulfonate (PTS) crystals. The parabolic law can closely describe the nonlinear interaction between the high-frequency Langmuir waves and the ion acoustic waves by ponderomotive forces [15, 16], in a region of reduced plasma density, and the nonlinear interaction between Langmuir waves and electrons. In addition, CQ was experimentally proposed as an empirical description of special semiconductor (e.g., AlGaAs, CdS, etc.) waveguides and semiconductor-doped glasses, particularly for the *CdSxSe*<sup>1</sup>�*<sup>x</sup>*-doped glass, which exhibit a significant fifthorder susceptibilities *χ*ð Þ<sup>5</sup> as experimentally reported earlier [17, 18]. Moreover, using high laser intensity, the saturation of nonlinearity has been established experimentally in many materials such as nonlinear organic polymers, semiconductor-doped glasses, and so on, which have the property that their absorption coefficient decreases [19]. More generally, a self-defocusing *χ*ð Þ<sup>5</sup> usually accounts for the saturation of *χ*ð Þ<sup>3</sup> . In recent years, many influential works have devoted to construct exact analyt-

ical solutions of CQNLSE, such as the pioneering work of Serkin et al. [20]. In particular, Dai et al. [21–25] obtained exact self-similar solutions (similaritons), their nonlinear tunneling effects of the generalized CQNLSE, and their higherdimensional forms with spatially inhomogeneous group velocity dispersion, cubic-

nanoplatelet colloids using a femtosecond laser [26], an extension of nonlinear Schrödinger equation including the cubic-quintic-septic nonlinearity was used to model the propagation of spatial solitons. In [27], for example, the authors performed numerical calculations based on higher-order nonlinearity parameters including seventh-order susceptibility *χ*ð Þ<sup>7</sup> (a chalcogenide glass is an example). This seeds several motivations to discover new features of solitons with combined effects of higher-order nonlinear parameters. In this regard, Houria et al. [28] constructed dark spatial solitary waves in a cubic-quintic-septic-nonlinear medium, with a profile in a

> *=*3

aspects of filamentation in gases and condensed matters [30–33].

solitary pulses for a derivative nonical-NLS equation on a CW background [29]. It is obvious to notice that the contributions of the higher-order nonlinearities can give way to generate stable solitons in homogeneous isotropic media and influence many

". They have also investigated chirped

Since the measurement of third-, fifth-, and seventh-order nonlinearities of silver

quintic nonlinearity, and amplification or attenuation.

functional form given in terms of "*sech*<sup>2</sup>

**212**

The dynamics of (1 + 1)-dimensional (one spatial and one temporal variables) spatial optical solitons is the well-known nonlinear Schrödinger equation. If we consider the higher-order effects, an extended model is required, and the propagation of optical pulses through the highly nonlinear waveguides can be described by the CQSNNLSE:

$$E\_x = ia\_1E\_{tt} + ia\_2|E|^2E + ia\_3|E|^4E + ia\_4|E|^6E + ia\_5|E|^8E,\tag{1}$$

where *E z*ð Þ , *t* is the slowly varying envelope of the electric field, the subscripts *z* and *t* are the spatial and temporal partial derivatives in the frame moving with the pulsed solutions, *α*<sup>1</sup> is the parameter of diffraction or dispersion, and *α*2, *α*3, *α*4, and *α*<sup>5</sup> are the cubic, quintic, septic, and nonical nonlinear terms, respectively. This model is relevant to some applications in which higher-order nonlinearities are important.

For example, Eq. (1) with *<sup>α</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> 2 , *α*<sup>4</sup> ¼ 1, and *α*<sup>5</sup> ¼ 0 was used to study numerically the stability conditions of one-dimensional spatial solitons [38]. Recently, Eq. (1) with *α*<sup>5</sup> ¼ 0 was analyzed for systems that are valid for several types of septic nonlinear materials [28]. Here, we consider arbitrary parameters *α <sup>j</sup>*ð Þ *j* ¼ 1, 2, 3, 4, 5 for the sake of a general analysis that is valid for several types of nonical media.

To obtain the exact analytic optical solitary-wave solutions of Eq. (1), we can employ the following transformation:

$$E(\mathbf{z}, \mathbf{t}\_{\cdot}) = \theta(\mathbf{t} + \beta \mathbf{z}) e^{i(k\mathbf{z} - \alpha t)} = \theta(\xi) e^{i(k\mathbf{z} - \alpha t)}.\tag{2}$$

Here, *θ ζ*ð Þ is a real function and *β* is a real constant to be determined.

Upon substituting Eq. (2) into Eq. (1) and separating the real and imaginary parts, one obtains

$$
\mathfrak{P} = \mathfrak{A}a\_1 a,\tag{3}
$$

$$
\theta\_{\xi\xi} = \frac{k + a\_1 \alpha^2}{a\_1} \theta - \frac{a\_2}{a\_1} \theta^3 - \frac{a\_3}{a\_1} \theta^5 - \frac{a\_4}{a\_1} \theta^7 - \frac{a\_5}{a\_1} \theta^9,\tag{4}
$$

Eq. (4) represents the evolution of an anharmonic oscillator with an effective potential energy *V* [28] defined by

$$\mathbf{V}(\theta) = -\frac{k + a\_1 \alpha^2}{2a\_1} \theta^2 + \frac{a\_2}{4a\_1} \theta^4 + \frac{a\_3}{6a\_1} \theta^6 + \frac{a\_4}{8a\_1} \theta^8 + \frac{a\_5}{10a\_1} \theta^{10}.\tag{5}$$

Integrating Eq. (4) yields

$$\left(\theta\_{\xi}\right)^{2} = a\_{1}\theta^{2} - a\_{2}\theta^{4} - a\_{3}\theta^{6} - a\_{4}\theta^{8} - a\_{5}\theta^{10} + 2\xi,\tag{6}$$

where

$$a\_1 = \frac{k + a\_1 \alpha^2}{2a\_1}, a\_n = \frac{a\_n}{na\_1} (n = 2, 3, 4, 5), \tag{7}$$

and *ξ* is the constant of integration, which can represent the energy of the anharmonic oscillator [39].

In order to get the exact soliton solutions, we first rewrite Eq. (6) in a simplified form by using transformation:

$$
\theta(\xi) = \mathfrak{u}^{\natural\_2}(\xi). \tag{8}
$$

*Ab* ¼

*Bright, Dark, and Kink Solitary Waves in a Cubic-Quintic-Septic-Nonical Medium*

8

>>>>>>>>>>>>>><

>>>>>>>>>>>>>>:

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *a***3 <sup>2</sup>**þ**4***a***1***a***<sup>5</sup>** *a***3 2**

The dark solitary solutions of Eq. (9) take the form [40]:

with parametric conditions

*DOI: http://dx.doi.org/10.5772/intechopen.92819*

**1** þ

subject to the parametric conditions

**2***a***1***Nd a***3**

8 ><

>:

**3.2 Dark solitary-wave solutions**

q

are of the form:

*Eb*ð Þ¼ *z*,*t*

*Ad*, *αd*, and energy *ξ*:

form:

**215**

*Ed*ð Þ¼ *z*,*t*

*Nb* ¼

*a***2***a***<sup>3</sup>** þ **4***a***1***a***<sup>4</sup>** ¼ **0**, *a***<sup>1</sup>** >**0**, *a***<sup>3</sup>** >**0**, *a***<sup>3</sup>**

**2***a***<sup>1</sup>** *a***3**

*cosh* ffiffiffiffiffi *a***1**

*ud*ð Þ¼ *<sup>ζ</sup> Ad sinh* ð Þ *<sup>α</sup>d<sup>ζ</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi **<sup>1</sup>** <sup>þ</sup> *Nd sinh* **<sup>2</sup>**

Here *Nd* is a real constant supposed to be positive. Real parameters *Ad* and *α<sup>d</sup>* are related to the amplitude and pulse width of the dark wave profiles, respectively. By substituting the ansatz Eq. (14) into Eq. (9), we get the unknown parameters

r

The exact dark solitary-wave solutions on a CW background of Eq. (1) are of the

*a***1**

ffiffiffiffiffiffiffiffiffiffiffiffiffi **2***a***1***Nd a***3**

*Ad* ¼

8 >>>>>><

>>>>>>:

**2**ð*a***2***a***<sup>3</sup>** þ **2***a***1***a***4**Þ � *a***<sup>3</sup>** ¼ **0**, *a***<sup>1</sup>** > **0**, *a***<sup>3</sup>** >**0**, *a***<sup>3</sup>**

*sinh* **<sup>2</sup>** ffiffiffiffiffi *a***1** <sup>p</sup>**<sup>2</sup>** ð Þ *<sup>t</sup>* <sup>þ</sup> *<sup>β</sup><sup>z</sup>* � �

<sup>p</sup>**<sup>2</sup>** ð Þ *<sup>t</sup>* <sup>þ</sup> **<sup>2</sup>***α***1***ω<sup>z</sup>* � � ( )**<sup>1</sup>**

**<sup>1</sup>** <sup>þ</sup> *Nd sinh* **<sup>2</sup>** ffiffiffiffiffi

*α<sup>d</sup>* ¼ ffiffiffiffiffi *a***1** p**2**

*<sup>ξ</sup>* <sup>¼</sup> *<sup>a</sup>***<sup>1</sup> 2***a***<sup>3</sup>**

*<sup>ξ</sup>* <sup>¼</sup> *<sup>a</sup>***1***a***<sup>2</sup> 2***a***<sup>3</sup>**

*α<sup>b</sup>* ¼ ffiffiffiffiffi *a***1** p**4**

ffiffiffiffiffiffiffi **2***a***<sup>1</sup>** *a***3**

*a***3**

Thus, the exact bright solitary-wave solutions on a CW background of Eq. (1)

<sup>p</sup>**<sup>4</sup>** ð Þ *<sup>t</sup>* <sup>þ</sup> **<sup>2</sup>***α***1***ω<sup>z</sup>* � �

9 >=

**1 4**

>;

<sup>q</sup> *:* (14)

, (15)

**<sup>2</sup>** <sup>þ</sup> **<sup>4</sup>***a***1***a***<sup>5</sup>** <sup>&</sup>gt; **<sup>0</sup>***:* (16)

� *<sup>e</sup>i kz* ð Þ �*ω<sup>t</sup> :* (17)

**4**

ð Þ *αdζ*

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

**<sup>2</sup>** <sup>þ</sup> **<sup>4</sup>***a***1***a***<sup>5</sup>** *a***3 2**

, (11)

**<sup>2</sup>** <sup>þ</sup> **<sup>4</sup>***a***1***a***<sup>5</sup>** <sup>&</sup>gt;**0***:* (12)

� *<sup>e</sup>i kz* ð Þ �*ω<sup>t</sup> :* (13)

r

By substituting Eq. (8) into Eq. (6), we obtain a new auxiliary equation possessing a sixth-degree nonlinear term:

$$\frac{1}{4} \left( u\_{\xi} \right)^{2} = a\_{1}u^{2} - a\_{2}u^{3} - a\_{3}u^{4} - a\_{4}u^{5} - a\_{5}u^{6} + 2\xi. \tag{9}$$

To solve Eq. (9), we will employ three types of localized solutions named bright, dark, and kink solitons. In the following, we solve Eq. (9) by using appropriate ansatz and obtain alternative types of solitary-wave solutions on a CW background and investigate parameter domains in which these optical spatial solitary waves exist.

### **3. Exact solitary-wave solutions**

In this section, we find bright-, dark-, and kink-solitary-wave localized solutions of Eq. (9), by using a special ansatz:

#### **3.1 Bright solitary-wave solutions**

The bright solitary solutions of Eq. (9) have the form:

$$\mu\_b(\xi) = \frac{A\_b}{\sqrt{\mathbf{1} + N\_b \cosh \left(a\_b \xi\right)}},\tag{10}$$

where *Ab*, *Nb*, and *α<sup>b</sup>* are real constants which represent wave parameters (*Ab* and *α<sup>b</sup>* related to the amplitude and pulse width of the bright wave profiles, respectively) to be determined by the physical coefficients of the model.

Substituting the ansatz Eq. (10) into Eq. (9), we obtain the unknown parameters *Ab*, *Nb*, *αb*, and energy *ξ*:

*Bright, Dark, and Kink Solitary Waves in a Cubic-Quintic-Septic-Nonical Medium DOI: http://dx.doi.org/10.5772/intechopen.92819*

$$\begin{cases} \mathbf{A}\_b = \sqrt{\frac{2a\_1}{a\_3}} \\\\ \mathbf{a}\_b = \sqrt[4]{a\_1} \\\\ \mathbf{N}\_b = \sqrt{\frac{a\_3^2 + 4a\_1a\_5}{a\_3^2}} \\\\ \xi = \frac{a\_1a\_2}{2a\_3} \end{cases} \tag{11}$$

with parametric conditions

Eq. (4) represents the evolution of an anharmonic oscillator with an effective

*<sup>θ</sup>***<sup>4</sup>** <sup>þ</sup>

, *an* <sup>¼</sup> *<sup>α</sup><sup>n</sup> nα***<sup>1</sup>**

and *ξ* is the constant of integration, which can represent the energy of the

*θ ζ*ð Þ¼ *<sup>u</sup>***<sup>1</sup>**

By substituting Eq. (8) into Eq. (6), we obtain a new auxiliary equation

dark, and kink solitons. In the following, we solve Eq. (9) by using appropriate ansatz and obtain alternative types of solitary-wave solutions on a CW background and investigate parameter domains in which these optical spatial solitary waves

In order to get the exact soliton solutions, we first rewrite Eq. (6) in a simplified

*=*

To solve Eq. (9), we will employ three types of localized solutions named bright,

In this section, we find bright-, dark-, and kink-solitary-wave localized solutions

*ub*ð Þ¼ *<sup>ζ</sup> Ab* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

where *Ab*, *Nb*, and *α<sup>b</sup>* are real constants which represent wave parameters (*Ab* and *α<sup>b</sup>* related to the amplitude and pulse width of the bright wave profiles,

Substituting the ansatz Eq. (10) into Eq. (9), we obtain the unknown parameters

respectively) to be determined by the physical coefficients of the model.

**<sup>1</sup>** <sup>þ</sup> *Nb cosh* ð Þ *<sup>α</sup>b<sup>ζ</sup>* <sup>p</sup> , (10)

� �**<sup>2</sup>** <sup>¼</sup> *<sup>a</sup>***1***u***<sup>2</sup>** � *<sup>a</sup>***2***u***<sup>3</sup>** � *<sup>a</sup>***3***u***<sup>4</sup>** � *<sup>a</sup>***4***u***<sup>5</sup>** � *<sup>a</sup>***5***u***<sup>6</sup>** <sup>þ</sup> **<sup>2</sup>***ξ:* (9)

*α***3 6***α***<sup>1</sup>**

*<sup>θ</sup>***<sup>6</sup>** <sup>þ</sup>

� �**<sup>2</sup>** <sup>¼</sup> *<sup>a</sup>***1***θ***<sup>2</sup>** � *<sup>a</sup>***2***θ***<sup>4</sup>** � *<sup>a</sup>***3***θ***<sup>6</sup>** � *<sup>a</sup>***4***θ***<sup>8</sup>** � *<sup>a</sup>***5***θ***<sup>10</sup>** <sup>þ</sup> **<sup>2</sup>***ξ*, (6)

*α***4 8***α***<sup>1</sup>** *<sup>θ</sup>***<sup>8</sup>** <sup>þ</sup>

*α***5 10***α***<sup>1</sup>**

ð Þ *n* ¼ **2**, **3**, **4**, **5** , (7)

**<sup>2</sup>**ð Þ*ζ :* (8)

*θ***<sup>10</sup>***:* (5)

potential energy *V* [28] defined by

Integrating Eq. (4) yields

anharmonic oscillator [39].

form by using transformation:

possessing a sixth-degree nonlinear term:

**1 <sup>4</sup>** *<sup>u</sup><sup>ζ</sup>*

**3. Exact solitary-wave solutions**

of Eq. (9), by using a special ansatz:

**3.1 Bright solitary-wave solutions**

*Ab*, *Nb*, *αb*, and energy *ξ*:

**214**

The bright solitary solutions of Eq. (9) have the form:

where

exist.

*<sup>V</sup>*ð Þ¼� *<sup>θ</sup> <sup>k</sup>* <sup>þ</sup> *<sup>α</sup>***1***ω***<sup>2</sup>**

*Nonlinear Optics - From Solitons to Similaritons*

*θζ*

**2***α***<sup>1</sup>**

*<sup>a</sup>***<sup>1</sup>** <sup>¼</sup> *<sup>k</sup>* <sup>þ</sup> *<sup>α</sup>***1***ω***<sup>2</sup> 2***α***<sup>1</sup>**

*<sup>θ</sup>***<sup>2</sup>** <sup>þ</sup>

*α***2 4***α***<sup>1</sup>**

$$a\_2 a\_3 + 4a\_1 a\_4 = \mathbf{0}, a\_1 > \mathbf{0}, a\_3 > \mathbf{0}, a\_3^2 + 4a\_1 a\_5 > \mathbf{0}.\tag{12}$$

Thus, the exact bright solitary-wave solutions on a CW background of Eq. (1) are of the form:

$$E\_b(\mathbf{z}, t) = \left\{ \frac{\frac{2a\_1}{a\_3}}{\mathbf{1} + \sqrt{\frac{a\_3^2 + 4a\_1a\_3}{a\_3^2}} \cosh\left[\sqrt[4]{a\_1}(t + 2a\_1oa)\right]} \right\}^{\frac{1}{4}} \times e^{i(kx - at)}.\tag{13}$$

#### **3.2 Dark solitary-wave solutions**

The dark solitary solutions of Eq. (9) take the form [40]:

$$u\_d(\xi) = \frac{A\_d \sinh\left(a\_d \xi\right)}{\sqrt{\mathbf{1} + \mathbf{N}\_d \sinh^2\left(a\_d \xi\right)}}.\tag{14}$$

Here *Nd* is a real constant supposed to be positive. Real parameters *Ad* and *α<sup>d</sup>*

are related to the amplitude and pulse width of the dark wave profiles, respectively. By substituting the ansatz Eq. (14) into Eq. (9), we get the unknown parameters *Ad*, *αd*, and energy *ξ*:

$$\begin{cases} A\_d = \sqrt{\frac{2a\_1 N\_d}{a\_3}} \\\\ a\_d = \sqrt[3]{a\_1} \\\\ \xi = \frac{a\_1}{2a\_3} \end{cases},\tag{15}$$

subject to the parametric conditions

$$\mathcal{Z}(a\_2a\_3 + \mathcal{Z}a\_1a\_4) - a\_3 = \mathbf{0}, a\_1 > \mathbf{0}, a\_3 > \mathbf{0}, a\_3^{-2} + 4a\_1a\_5 > \mathbf{0}.\tag{16}$$

The exact dark solitary-wave solutions on a CW background of Eq. (1) are of the form:

$$E\_d(\mathbf{z}, t) = \left\{ \frac{2a\_1 N\_d}{a\_3} \frac{\sinh^2 \left[ \sqrt[3]{a\_1} (t + \beta \mathbf{z}) \right]}{\mathbf{1} + N\_d \sinh^2 \left[ \sqrt[3]{a\_1} (t + 2a\_1 \alpha \mathbf{z}) \right]} \right\}^{\frac{1}{4}} \times e^{i(k\mathbf{z} - \alpha t)}. \tag{17}$$

### **3.3 Kink solitary-wave solutions**

The kink solitary solutions of Eq. (9) are in the following form:

$$
\mathfrak{u}\_k(\xi) = \mathbf{A}\_k \sqrt{\mathbf{1} + \tanh(a\_k \xi)}, \tag{18}
$$

(with *α***<sup>4</sup>** ¼ **0**, *α***<sup>5</sup>** ¼ **0**) that is available in the current literature. As we can see from Eq. (21), the kink solitons exist only if *a***<sup>5</sup>** 6¼ **0**, consequently *α***<sup>5</sup>** 6¼ **0**; thus, we cannot

*Bright, Dark, and Kink Solitary Waves in a Cubic-Quintic-Septic-Nonical Medium*

One of the essential aspects of solitary waves is their stability on propagation, in particular their ability to propagate in a perturbed environment over an appreciable distance [41]. Unlike the conventional pulses of different forms, the solitons are relatively stable, even in an environment subjected to external perturbations.

The previous three exact solitary-wave solutions given by the expressions (13), (17), and (21) are sitting on a CW background, which may be subject to MI. If this phenomenon occurs, then the CW background will be quickly destroyed, which will inevitably cause the destruction of the soliton. It is therefore of paramount importance to verify whether the condition of the existence of the soliton can be compatible with the condition of the stability of the CW background. Since MI properties can be used to understand the different excitation patterns on a CW in nonlinear systems, in this section, we perform the standard linear stability analysis

*<sup>E</sup>***0**ð Þ¼ *<sup>z</sup>*,*<sup>t</sup>* ffiffiffiffiffi

is the nonlinear phase shift induced by self-phase modulation and non-Kerr quintic-

exhibiting optical nonlinearities up to the ninth order. A perturbed nonlinear back-

*P***0** <sup>p</sup> <sup>þ</sup> *a z*ð Þ ,*<sup>t</sup>*

where *a z*ð Þ , *t* is a small perturbation field which is given by collecting the Fourier

*a*<sup>þ</sup> and *a*� are much less than the background amplitude *P*0, and *Ω* represents

the perturbed frequency. Here, the complex field |*a z*ð Þ , *t* | ≪ *P*0. Thus, if the perturbed field grows exponentially, the steady state (CW) becomes unstable. Inserting the expression of a perturbed nonlinear background Eq. (23) into Eq. (1), with respect to Eq. (24), we obtain after linearization the following dispersion

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> <sup>þ</sup> <sup>6</sup>*α*4*P*<sup>0</sup>

depending on the sign of *α*<sup>1</sup> ½ � *sgn* ð Þ¼þ *α*<sup>1</sup> 1, for *α*<sup>1</sup> > 0, and *sgn* ð Þ¼� *α*<sup>1</sup> 1, for *α*<sup>1</sup> <0 . The dispersion relation (25) shows that the steady-state stability depends critically on whether the light experiences normal or anomalous GVD inside the fiber. In the case of normal GVD (*α*<sup>1</sup> < 0), the wave number *K* is real for all *Ω*, and the steady state is stable against small perturbations. By contrast, in the case of anomalous

*K* ¼ |*α***1**||*Ω*|

2*α*2*P*<sup>0</sup> þ 4*α*3*P*<sup>0</sup>

in the system modeled by Eq. (1), where *ϕnl* ¼ *P***<sup>0</sup>** *α***<sup>2</sup>** þ *α***3***P***<sup>0</sup>** þ *α***4***P***<sup>0</sup>**

septic-nonical nonlinear terms, *P***<sup>0</sup>** being the initial power inside a medium

ground plane-wave field for the CQSNNLSE (Eq. (1)) can be written as

*E z*ð Þ¼ ,*<sup>t</sup>* ffiffiffiffiffi

*P***0**

p *eiϕnl*, (22)

h i*eiϕnl*, (23)

*a z*ð Þ¼ ,*<sup>t</sup> <sup>a</sup>*þ*ei Kz* ð Þ �*Ω<sup>t</sup>* <sup>þ</sup> *<sup>a</sup>*�*e*�*i Kz* ð Þ �*Ω<sup>t</sup>* , (24)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>Ω</sup>***<sup>2</sup>** � *sgn* ð Þ *<sup>α</sup>***<sup>1</sup>** *<sup>Ω</sup><sup>c</sup>*

<sup>3</sup> <sup>þ</sup> <sup>8</sup>*α*5*P*<sup>0</sup> <sup>4</sup> <sup>p</sup> and *sgn* ð Þ¼� *<sup>α</sup>*<sup>1</sup> <sup>1</sup>

**2**

**<sup>3</sup>** � �*z*

, (25)

**<sup>2</sup>** <sup>þ</sup> *<sup>α</sup>***5***P***<sup>0</sup>**

plot the corresponding CQNLS kink solution.

*DOI: http://dx.doi.org/10.5772/intechopen.92819*

[9, 34] on a generic CW:

modes as

relation:

**217**

where *<sup>Ω</sup><sup>c</sup>* <sup>¼</sup> <sup>1</sup>

ffiffiffiffiffi *α*1 p

**4. Modulational instability of the CW background**

where *Ak* and *α<sup>k</sup>* are real parameters related to the amplitude and pulse width of the kink wave profiles, respectively.

Substituting Eq. (18) into Eq. (9), we get

$$\begin{cases} \mathcal{A}\_k = \sqrt{-\frac{a\_3}{4a\_5}}\\ a\_k = \sqrt{-\frac{a\_3^2}{4a\_5}}, \\ \mathfrak{k} = -\frac{a\_2^2}{4a\_4} \end{cases} \tag{19}$$

under the parametric conditions

$$2a\_2a\_5 - a\_3a\_4 = \mathbf{0}, a\_3^{-2} + 4a\_1a\_5 = \mathbf{0}, a\_3 > \mathbf{0}, a\_5 < \mathbf{0}.\tag{20}$$

Thus, the exact bright solitary-wave solutions on a CW background of Eq. (1) are of the form:

$$E\_k(\mathbf{z}, \mathbf{t}) = \left\{-\frac{a\_3}{4a\_5} \left[\mathbf{1} + \tanh\left(\sqrt{-\frac{a\_3^2}{4a\_5}}(\mathbf{t} + 2a\_1a\mathbf{z})\right)\right] \right\}^{\frac{1}{4}} \times \mathbf{e}^{i(kx - \alpha t)}.\tag{21}$$

The previous three exact solitary-wave solutions (13), (17), and (21) exist for the governing nonical-NLS model due to a balance among diffraction (or dispersion) and competing cubic-quintic-septic-nonical nonlinearities. For better insight, we plot in **Figure 1** the intensity profile on top of the related first two exact solution solitons named bright and dark, corresponding to the CQNLS models

**Figure 1.**

*Intensity |Ej(z, t)|<sup>2</sup> distribution of the (a) bright and (b) dark solitons given by Eqs. (14) and (17), respectively, with the parameter values corresponding to CQNLS models as α*<sup>1</sup> ¼ 0*:*5*, α*<sup>2</sup> ¼ 0*, α*<sup>3</sup> ¼ 1*, α*<sup>4</sup> ¼ 0*, α*<sup>5</sup> ¼ 0*, k* ¼ 1*, and ω* ¼ 1*.*

*Bright, Dark, and Kink Solitary Waves in a Cubic-Quintic-Septic-Nonical Medium DOI: http://dx.doi.org/10.5772/intechopen.92819*

(with *α***<sup>4</sup>** ¼ **0**, *α***<sup>5</sup>** ¼ **0**) that is available in the current literature. As we can see from Eq. (21), the kink solitons exist only if *a***<sup>5</sup>** 6¼ **0**, consequently *α***<sup>5</sup>** 6¼ **0**; thus, we cannot plot the corresponding CQNLS kink solution.

## **4. Modulational instability of the CW background**

One of the essential aspects of solitary waves is their stability on propagation, in particular their ability to propagate in a perturbed environment over an appreciable distance [41]. Unlike the conventional pulses of different forms, the solitons are relatively stable, even in an environment subjected to external perturbations.

The previous three exact solitary-wave solutions given by the expressions (13), (17), and (21) are sitting on a CW background, which may be subject to MI. If this phenomenon occurs, then the CW background will be quickly destroyed, which will inevitably cause the destruction of the soliton. It is therefore of paramount importance to verify whether the condition of the existence of the soliton can be compatible with the condition of the stability of the CW background. Since MI properties can be used to understand the different excitation patterns on a CW in nonlinear systems, in this section, we perform the standard linear stability analysis [9, 34] on a generic CW:

$$\mathbf{E\_0(z,t)} = \sqrt{P\_0} \mathbf{e^{i\phi\_{ul}}},\tag{22}$$

in the system modeled by Eq. (1), where *ϕnl* ¼ *P***<sup>0</sup>** *α***<sup>2</sup>** þ *α***3***P***<sup>0</sup>** þ *α***4***P***<sup>0</sup> <sup>2</sup>** <sup>þ</sup> *<sup>α</sup>***5***P***<sup>0</sup> <sup>3</sup>** � �*z* is the nonlinear phase shift induced by self-phase modulation and non-Kerr quinticseptic-nonical nonlinear terms, *P***<sup>0</sup>** being the initial power inside a medium exhibiting optical nonlinearities up to the ninth order. A perturbed nonlinear background plane-wave field for the CQSNNLSE (Eq. (1)) can be written as

$$E(\mathbf{z}, \mathbf{t}) = \left[\sqrt{P\_0} + \mathfrak{a}(\mathbf{z}, \mathbf{t})\right] \mathbf{e}^{i\phi\_{nl}},\tag{23}$$

where *a z*ð Þ , *t* is a small perturbation field which is given by collecting the Fourier modes as

$$\mathfrak{a}(\mathfrak{z}, \mathfrak{t}) = \mathfrak{a}\_{+} \mathfrak{e}^{i(\mathbf{K}\mathfrak{z} - \mathfrak{A}\mathfrak{t})} + \mathfrak{a}\_{-} \mathfrak{e}^{-i(\mathbf{K}\mathfrak{z} - \mathfrak{A}\mathfrak{t})},\tag{24}$$

*a*<sup>þ</sup> and *a*� are much less than the background amplitude *P*0, and *Ω* represents the perturbed frequency. Here, the complex field |*a z*ð Þ , *t* | ≪ *P*0. Thus, if the perturbed field grows exponentially, the steady state (CW) becomes unstable. Inserting the expression of a perturbed nonlinear background Eq. (23) into Eq. (1), with respect to Eq. (24), we obtain after linearization the following dispersion relation:

$$K = |a\_1||\mathcal{Q}|\sqrt{\mathcal{Q}^2 - \text{sgn}\,(a\_1)\mathcal{Q}\_c}^2,\tag{25}$$

where *<sup>Ω</sup><sup>c</sup>* <sup>¼</sup> <sup>1</sup> ffiffiffiffiffi *α*1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2*α*2*P*<sup>0</sup> þ 4*α*3*P*<sup>0</sup> <sup>2</sup> <sup>þ</sup> <sup>6</sup>*α*4*P*<sup>0</sup> <sup>3</sup> <sup>þ</sup> <sup>8</sup>*α*5*P*<sup>0</sup> <sup>4</sup> <sup>p</sup> and *sgn* ð Þ¼� *<sup>α</sup>*<sup>1</sup> <sup>1</sup> depending on the sign of *α*<sup>1</sup> ½ � *sgn* ð Þ¼þ *α*<sup>1</sup> 1, for *α*<sup>1</sup> > 0, and *sgn* ð Þ¼� *α*<sup>1</sup> 1, for *α*<sup>1</sup> <0 . The dispersion relation (25) shows that the steady-state stability depends critically on whether the light experiences normal or anomalous GVD inside the fiber. In the case of normal GVD (*α*<sup>1</sup> < 0), the wave number *K* is real for all *Ω*, and the steady state is stable against small perturbations. By contrast, in the case of anomalous

**3.3 Kink solitary-wave solutions**

*Nonlinear Optics - From Solitons to Similaritons*

the kink wave profiles, respectively.

under the parametric conditions

*a***3 4***a***<sup>5</sup>**

are of the form:

**Figure 1.**

**216**

*α*<sup>5</sup> ¼ 0*, k* ¼ 1*, and ω* ¼ 1*.*

*Ek*ð Þ¼ � *z*,*t*

**2***a***2***a***<sup>5</sup>** � *a***3***a***<sup>4</sup>** ¼ **0**, *a***<sup>3</sup>**

**1** þ *tanh*

Substituting Eq. (18) into Eq. (9), we get

The kink solitary solutions of Eq. (9) are in the following form:

*Ak* ¼

*α<sup>k</sup>* ¼

*<sup>ξ</sup>* ¼ � *<sup>a</sup>***<sup>2</sup>**

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffi � *<sup>a</sup>***<sup>3</sup> 4***a***<sup>5</sup>**

ffiffiffiffiffiffiffiffiffiffiffiffi � *a***3 2 4***a***<sup>5</sup>**

**2 4***a***<sup>4</sup>**

where *Ak* and *α<sup>k</sup>* are real parameters related to the amplitude and pulse width of

r

s

Thus, the exact bright solitary-wave solutions on a CW background of Eq. (1)

ffiffiffiffiffiffiffiffiffiffiffiffi � *a***3 2 4***a***<sup>5</sup>**

The previous three exact solitary-wave solutions (13), (17), and (21) exist for the governing nonical-NLS model due to a balance among diffraction (or dispersion) and competing cubic-quintic-septic-nonical nonlinearities. For better insight, we plot in **Figure 1** the intensity profile on top of the related first two exact solution solitons named bright and dark, corresponding to the CQNLS models

ð Þ *t* þ **2***α***1***ωz*

s

( ) " # ! **<sup>1</sup>**

*Intensity |Ej(z, t)|<sup>2</sup> distribution of the (a) bright and (b) dark solitons given by Eqs. (14) and (17), respectively, with the parameter values corresponding to CQNLS models as α*<sup>1</sup> ¼ 0*:*5*, α*<sup>2</sup> ¼ 0*, α*<sup>3</sup> ¼ 1*, α*<sup>4</sup> ¼ 0*,*

**<sup>1</sup>** <sup>þ</sup> *tanh* ð Þ *<sup>α</sup>k<sup>ζ</sup>* <sup>p</sup> , (18)

**<sup>2</sup>** <sup>þ</sup> **<sup>4</sup>***a***1***a***<sup>5</sup>** <sup>¼</sup> **<sup>0</sup>**, *<sup>a</sup>***<sup>3</sup>** <sup>&</sup>gt;**0**, *<sup>a</sup>***<sup>5</sup>** <sup>&</sup>lt;**0***:* (20)

**4**

� *<sup>e</sup>i kz* ð Þ �*ω<sup>t</sup> :* (21)

, (19)

*uk*ð Þ¼ *ζ Ak*

8

>>>>>>>><

>>>>>>>>:

GVD (*α*<sup>1</sup> > 0), *K* becomes imaginary for *Ω* < *Ωc*, and the perturbation *a z*ð Þ , *t* grows exponentially with *z*. As a result, the CW solution *E*<sup>0</sup> is inherently unstable for *α*<sup>1</sup> >0. This instability is referred to as modulation instability because it leads to a spontaneous temporal modulation of the CW beam and transforms it into a pulse train. Similar instabilities occur in many other nonlinear systems and are often called self-pulsing instabilities [9, 42–45]. Then, by setting *sgn* ð Þ¼ *α*<sup>1</sup> 1, one can obtain the MI gain *G* ¼ 2 *Im K*ð Þ, where the factor 2 convert *G* to power gain. The gain exists only if for |*Ω*|< *Ω<sup>c</sup>* and is given by

$$\mathbf{G}(\mathfrak{Q}) = \mathfrak{Q}|\mathfrak{a\_1}\mathfrak{Q}|\sqrt{\mathfrak{Q\_c}^2 - \mathfrak{Q}^2}. \tag{26}$$

We can observe that the OMF increases (respectively decreases) with the

*Bright, Dark, and Kink Solitary Waves in a Cubic-Quintic-Septic-Nonical Medium*

*Variation of the MI gain G as a function of the nonic nonlinearity α*5*, with the same parameter values as in*

*Variation of the MI gain G km*�<sup>1</sup> ð Þ *as a function of frequency <sup>Ω</sup>* ð Þ *Hz , at a four-power level P*<sup>0</sup> *for an optical fiber. The other parameters are <sup>α</sup>*<sup>5</sup> <sup>¼</sup> <sup>0</sup>*:*5*ps*<sup>2</sup>*=km*, *<sup>α</sup>*<sup>2</sup> <sup>¼</sup> <sup>2736</sup>*W*�<sup>1</sup>*=km*, *<sup>α</sup>*<sup>3</sup> <sup>¼</sup> <sup>2</sup>*:*63*W*�<sup>2</sup>*=km*,

**Figure 3** shows the variation of MI gain as a function of the nonic nonlinearity *α***5**. The MI gain increases with the decreasing nonic nonlinearity. In **Figure 4**, as the

increasing *α***<sup>1</sup>** *<* **0** (respectively with the increasing *α***<sup>1</sup>** *>* **0**).

*DOI: http://dx.doi.org/10.5772/intechopen.92819*

input power increases, the maximum gain also increases.

**Figure 3.**

*Figure 2.*

**Figure 4.**

**219**

*<sup>α</sup>*<sup>4</sup> ¼ �9*:*<sup>12</sup> � <sup>10</sup>�<sup>4</sup>*W*�<sup>3</sup>*=km*, *<sup>α</sup>*<sup>5</sup> <sup>¼</sup> <sup>0</sup>*:*5*W*�<sup>4</sup>*=km.*

The gain attains its peak values when the modulated frequency reaches its optimum value, i.e., its optimum modulation frequency (OMF). The OMF corresponding to the gain spectrum (26) is given by

$$
\mathfrak{A}\_{\text{op}} = \pm \frac{\mathfrak{A}\_c}{\sqrt{2}},
\tag{27}
$$

and the peak value given by

$$\mathbf{G}\_{op} = \mathbf{G} \left( \mathfrak{Q}\_{op} \right) = |\mathfrak{a}\_1 \mathfrak{Q}\_c|^2 |. \tag{28}$$

In **Figure 2**, we have shown the variation of OMF, computed from Eq. (27) as a function of the GVD parameter (*α***1**). The parameter values we have used are given as [34]

$$\begin{aligned} P\_0 &= \mathbf{15W}, a\_2 = 2736 \,\mathrm{W}^{-1}/km, a\_3 = 2.63 \,\mathrm{W}^{-2}/km, \\ a\_4 &= -9.12 \times 10^{-4} \,\mathrm{W}^{-3}/km, a\_5 = \mathbf{0.5W}^{-4}/km. \end{aligned} \tag{29}$$

**Figure 2.** *Variation of optimum modulation frequency Ωop as a function of second-order dispersion α*1*.*

*Bright, Dark, and Kink Solitary Waves in a Cubic-Quintic-Septic-Nonical Medium DOI: http://dx.doi.org/10.5772/intechopen.92819*

We can observe that the OMF increases (respectively decreases) with the increasing *α***<sup>1</sup>** *<* **0** (respectively with the increasing *α***<sup>1</sup>** *>* **0**).

**Figure 3** shows the variation of MI gain as a function of the nonic nonlinearity *α***5**. The MI gain increases with the decreasing nonic nonlinearity. In **Figure 4**, as the input power increases, the maximum gain also increases.

**Figure 3.** *Variation of the MI gain G as a function of the nonic nonlinearity α*5*, with the same parameter values as in Figure 2.*

#### **Figure 4.**

GVD (*α*<sup>1</sup> > 0), *K* becomes imaginary for *Ω* < *Ωc*, and the perturbation *a z*ð Þ , *t* grows exponentially with *z*. As a result, the CW solution *E*<sup>0</sup> is inherently unstable for *α*<sup>1</sup> >0. This instability is referred to as modulation instability because it leads to a spontaneous temporal modulation of the CW beam and transforms it into a pulse train. Similar instabilities occur in many other nonlinear systems and are often called self-pulsing instabilities [9, 42–45]. Then, by setting *sgn* ð Þ¼ *α*<sup>1</sup> 1, one can obtain the MI gain *G* ¼ 2 *Im K*ð Þ, where the factor 2 convert *G* to power gain. The

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *Ωc* **<sup>2</sup>** � *<sup>Ω</sup>***<sup>2</sup>**

*:* (26)

p , (27)

**<sup>2</sup>**|*:* (28)

q

*Ωc* ffiffiffi **2**

� � <sup>¼</sup> <sup>|</sup>*α***1***Ω<sup>c</sup>*

*<sup>=</sup>km*, *<sup>α</sup>***<sup>3</sup>** <sup>¼</sup> **<sup>2</sup>***:***63***W*�**2***=km*,

*<sup>=</sup>km*, *<sup>α</sup>***<sup>5</sup>** <sup>¼</sup> **<sup>0</sup>***:***5***W*�**4***=km:* (29)

In **Figure 2**, we have shown the variation of OMF, computed from Eq. (27) as a function of the GVD parameter (*α***1**). The parameter values we have used are given

The gain attains its peak values when the modulated frequency reaches its optimum value, i.e., its optimum modulation frequency (OMF). The OMF

*Ωop* ¼ �

*Gop* ¼ *G Ωop*

*Variation of optimum modulation frequency Ωop as a function of second-order dispersion α*1*.*

*G*ð Þ¼ *Ω* **2**|*α***1***Ω*|

gain exists only if for |*Ω*|< *Ω<sup>c</sup>* and is given by

*Nonlinear Optics - From Solitons to Similaritons*

corresponding to the gain spectrum (26) is given by

*<sup>P</sup>***<sup>0</sup>** <sup>¼</sup> **<sup>15</sup>***W*, *<sup>α</sup>***<sup>2</sup>** <sup>¼</sup> **<sup>2736</sup>***W*�**<sup>1</sup>**

*<sup>α</sup>***<sup>4</sup>** ¼ �**9***:***<sup>12</sup>** � **<sup>10</sup>**�**4***W*�**<sup>3</sup>**

and the peak value given by

as [34]

**Figure 2.**

**218**

*Variation of the MI gain G km*�<sup>1</sup> ð Þ *as a function of frequency <sup>Ω</sup>* ð Þ *Hz , at a four-power level P*<sup>0</sup> *for an optical fiber. The other parameters are <sup>α</sup>*<sup>5</sup> <sup>¼</sup> <sup>0</sup>*:*5*ps*<sup>2</sup>*=km*, *<sup>α</sup>*<sup>2</sup> <sup>¼</sup> <sup>2736</sup>*W*�<sup>1</sup>*=km*, *<sup>α</sup>*<sup>3</sup> <sup>¼</sup> <sup>2</sup>*:*63*W*�<sup>2</sup>*=km*, *<sup>α</sup>*<sup>4</sup> ¼ �9*:*<sup>12</sup> � <sup>10</sup>�<sup>4</sup>*W*�<sup>3</sup>*=km*, *<sup>α</sup>*<sup>5</sup> <sup>¼</sup> <sup>0</sup>*:*5*W*�<sup>4</sup>*=km.*

#### **Figure 5.**

*Variation of the MI gain G km*�<sup>1</sup> ð Þ *as a function of frequency <sup>Ω</sup> and the GVD <sup>α</sup>*1*. The other parameters are <sup>P</sup>*<sup>0</sup> <sup>¼</sup> <sup>15</sup>*W*, *<sup>α</sup>*<sup>5</sup> ¼ �0*:*5*ps*<sup>2</sup>*=km*, *<sup>α</sup>*<sup>2</sup> <sup>¼</sup> <sup>2736</sup>*W*�<sup>1</sup>*=km*, *<sup>α</sup>*<sup>3</sup> <sup>¼</sup> <sup>2</sup>*:*63*W*�<sup>2</sup>*=km*, *<sup>α</sup>*<sup>4</sup> ¼ �9*:*<sup>12</sup> � <sup>10</sup>�<sup>4</sup>*W*�<sup>3</sup>*=km*, *<sup>α</sup>*<sup>5</sup> <sup>¼</sup> <sup>5</sup>*W*�<sup>4</sup>*=km.*

The MI gain spectrum in **Figure 5** is a constitutive of two symmetrical sidebands which stand symmetrically along the line *Ω* ¼ 0. The maximum gain is nil at the zero perturbation frequency *Ω* ¼ 0; thus, there is no instability at the zero perturbation frequency.

**Author details**

Mati Youssoufa<sup>1</sup>

Cameroon

Cameroon

Cameroon

**221**

mohdoufr@yahoo.fr

provided the original work is properly cited.

\*, Ousmanou Dafounansou1,2\* and Alidou Mohamadou1,3,4,5\*

1 Faculty of Science, Department of Physics, University of Douala, Douala,

*Bright, Dark, and Kink Solitary Waves in a Cubic-Quintic-Septic-Nonical Medium*

*DOI: http://dx.doi.org/10.5772/intechopen.92819*

2 Faculty of Science, Department of Physics, University of Maroua, Maroua,

3 National Advanced School of Engineering, University of Maroua, Maroua,

4 Max Planck Institute for the Physics of Complex Systems, Dreseden, Germany

5 The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

\*Address all correspondence to: mati.youss@yahoo.fr, dagaf10@yahoo.fr and

© 2020 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,

### **5. Conclusion**

In this chapter, we have investigated the higher-order nonlinear Schrödinger equation involving nonlinearity up to the ninth order. We have constructed exact solutions of this equation by means of a special ansatz. We showed the existence of a family of solitonic solutions: bright, dark, and kink solitons. The conditions on the physical parameters for the existence of this propagating envelope have also been reported. These conditions show a subtle balance among the diffraction or dispersion, Kerr nonlinearity, and quintic-septic-nonical non-Kerr nonlinearities, which has a profound implication to control the wave dynamics. Moreover, by employing Stuart and DiPrima's stability analysis method, an analytical expression for the MI gain has been obtained. The outcomes of the instability development depend on the nonlinearity and dispersion (or diffraction) parameters. Results may find straightforward applications in nonlinear optics, particularly in fiber-optical communication.

## **Conflict of interest**

The authors declare no conflict of interest.

*Bright, Dark, and Kink Solitary Waves in a Cubic-Quintic-Septic-Nonical Medium DOI: http://dx.doi.org/10.5772/intechopen.92819*

## **Author details**

The MI gain spectrum in **Figure 5** is a constitutive of two symmetrical sidebands which stand symmetrically along the line *Ω* ¼ 0. The maximum gain is nil at the zero perturbation frequency *Ω* ¼ 0; thus, there is no instability at the zero

*Variation of the MI gain G km*�<sup>1</sup> ð Þ *as a function of frequency <sup>Ω</sup> and the GVD <sup>α</sup>*1*. The other parameters are <sup>P</sup>*<sup>0</sup> <sup>¼</sup> <sup>15</sup>*W*, *<sup>α</sup>*<sup>5</sup> ¼ �0*:*5*ps*<sup>2</sup>*=km*, *<sup>α</sup>*<sup>2</sup> <sup>¼</sup> <sup>2736</sup>*W*�<sup>1</sup>*=km*, *<sup>α</sup>*<sup>3</sup> <sup>¼</sup> <sup>2</sup>*:*63*W*�<sup>2</sup>*=km*, *<sup>α</sup>*<sup>4</sup> ¼ �9*:*<sup>12</sup> � <sup>10</sup>�<sup>4</sup>*W*�<sup>3</sup>*=km*,

In this chapter, we have investigated the higher-order nonlinear Schrödinger equation involving nonlinearity up to the ninth order. We have constructed exact solutions of this equation by means of a special ansatz. We showed the existence of a family of solitonic solutions: bright, dark, and kink solitons. The conditions on the physical parameters for the existence of this propagating envelope have also been reported. These conditions show a subtle balance among the diffraction or dispersion, Kerr nonlinearity, and quintic-septic-nonical non-Kerr nonlinearities, which has a profound implication to control the wave dynamics. Moreover, by employing Stuart and DiPrima's stability analysis method, an analytical expression for the MI gain has

been obtained. The outcomes of the instability development depend on the

nonlinearity and dispersion (or diffraction) parameters. Results may find straightforward applications in nonlinear optics, particularly in fiber-optical communication.

perturbation frequency.

*Nonlinear Optics - From Solitons to Similaritons*

**Conflict of interest**

**220**

The authors declare no conflict of interest.

**5. Conclusion**

**Figure 5.**

*<sup>α</sup>*<sup>5</sup> <sup>¼</sup> <sup>5</sup>*W*�<sup>4</sup>*=km.*

Mati Youssoufa<sup>1</sup> \*, Ousmanou Dafounansou1,2\* and Alidou Mohamadou1,3,4,5\*

1 Faculty of Science, Department of Physics, University of Douala, Douala, Cameroon

2 Faculty of Science, Department of Physics, University of Maroua, Maroua, Cameroon

3 National Advanced School of Engineering, University of Maroua, Maroua, Cameroon

4 Max Planck Institute for the Physics of Complex Systems, Dreseden, Germany

5 The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

\*Address all correspondence to: mati.youss@yahoo.fr, dagaf10@yahoo.fr and mohdoufr@yahoo.fr

© 2020 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.

## **References**

[1] Hasegawa A, Tappert FD. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Applied Physics Letters. 1973;**23**:142

[2] Hasegawa A, Tappert FD. Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion. Applied Physics Letters. 1973;**23**:171

[3] Mollenauer LF, Stolen RH, Gordon JP. Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Physical Review Letters. 1980;**45**:1095

[4] Barthelemy A, Maneuf S, Froehly C. Soliton propagation and selfconfinement of laser-beams by Kerr optical nonlinearity. Optics Communication. 1985;**55**:201

[5] Hasegawa A, Kodama Y. Solitons in Optical Communications. Oxford: Oxford University Press; 1995

[6] Agrawal GP. Nonlinear Fiber Optics. New York: Academic Press; 2013

[7] Abdullaev F, Darmanyan S, Khabibullaev P. Optical Solitons. Berlin: Springer-Verlag; 1991

[8] Kivshar YS, Luther-Davies B. Dark optical solitons: Physics and applications. Physics Reports. 1998;**298**:81

[9] Kivshar YS, Agrawal GP. Optical Solitons: "From Fibers to Photonic Crystal". San Diego: Academic Press; 2003

[10] Agrawal GP, Headley C III. Kink solitons and optical shocks in dispersive nonlinear media. Physical Review A. 1992;**46**:1573

[11] Raju TS, Panigrahi PK. Self-similar propagation in a graded-index

nonlinear-fiber amplifier with an external source. Physical Review A. 2010;**81**:043820

[19] Stegeman GI, Stolen RH.

[20] Serkin VN, Belyaeva TL,

Bellingham: SPIE; 2001. p. 292

Letters. 2010;**35**:1437

**35**:2651

[21] Dai CQ, Wang YY, Zhang JF. Analytical spatiotemporal localizations for the generalized (3+1)-dimensional nonlinear Schrödinger equation. Optics

[22] Dai CQ, Zhang JF. Exact spatial similaritons and rogons in 2D gradedindex waveguides. Optics Letters. 2010;

[23] Dai CQ, Zhu SQ, Wang LL, Zhang JF. Exact spatial similaritons for the generalized (2+1)-dimensional nonlinear Schrödinger equation with distributed coefficients. Europhysics

[24] Dai CQ, Wang XG, Zhang JF. Nonautonomous spatiotemporal localized structures in the inhomogeneous optical fibers: interaction and control. Annals of Physics (NY). 2011;**326**:645

[25] Dai CQ, Yang Q, He JD, Wang YY. Nonlinear tunneling effect in the (2+1) dimensional cubic-quintic nonlinear Schrödinger equation with variable coefficients. The European Physical

[26] Jayabalan J, Singh A, Chari R, Khan S, Srivastava H, Oak SM.

Transient absorption and higher-order nonlinearities in silver nanoplatelets. Applied Physics Letters. 2009;**94**:

[27] Reyna AS, Jorge KC, de Araújo CB.

Two-dimensional solitons in a

Letters. 2010;**92**:24005

Journal. 2011;**D63**:141

181902

**223**

1989;**6**:652

Waveguides and fibers for nonlinear optics. Journal of the Optical Society of America B: Optical Physics.

*DOI: http://dx.doi.org/10.5772/intechopen.92819*

quintic-septimal medium. Physical

[28] Triki H, Porsezian K, Dinda PT, Grelu P. Dark spatial solitary waves in a cubic-quintic-septimal nonlinear medium. Physical Review A. 2017;**95**:

[29] Triki H, Porsezian K, Choudhuri A, Tchofo Dinda P. Chirped solitary pulses for a nonic nonlinear Schrödinger equation on a continuous-wave background. Physical Review A. 2016;

[30] Boyd RW, Lukishova SG, Shen YR, editors. Self-focusing: "Past and present (fundamentals and prospects)". In: Topics in Applied Physics. Vol. 114.

[31] Zeng J, Malomed BA. Bright solitons

in defocusing media with spatial modulation of the quintic nonlinearity. Physics Review. 2012;**E86**:036607

[32] Couairon A, Mysyrowicz A. Femtosecond filamentation in transparent media. Physics Reports.

[33] Liu W, Petit S, Becker A, Akцozbek N, Bowden CM, Chin SL. Intensity clamping of a femtosecond laser pulse in condensed matter. Optics

Communication. 2002;**202**:189

[35] Shukla PK, Rasmussen JJ.

[34] Mohamadou A, Latchio Tiofack CG, Kofané TC. Wave train generation of solitons in systems with higher-order nonlinearities. Physical Review E. 2010;

Modulational instability of short pulses in long optical fibers. Optics Letters.

[36] Potasek MJ. Modulation instability in an extended nonlinear Schrödinger equation. Optics Letters. 1987;**12**:921

Review A. 2014;**90**:063835

023837

*Bright, Dark, and Kink Solitary Waves in a Cubic-Quintic-Septic-Nonical Medium*

**93**:063810

2007;**441**:47

**82**:016601

1986;**11**:171

Berlin: Springer; 2009

Alexandrov IV, Melchor GM. Optical pulse and beam propagation III. In: Band YB, editor. SPIE Proceedings. Vol. 4271.

[12] Goyal A, Gupta R, Kumar CN, Raju TS. Chirped femtosecond solitons and double-kink solitons in the cubicquintic nonlinear Schrödinger equation with self-steepening and self-frequency shift. Physical Review A. 2011;**84**: 063830

[13] Porubov AV, Andrievsky BR. Kink and solitary waves may propagate together. Physical Review E. 2012;**85**: 046604

[14] Agrawal GP. Optical Solitons, Autosolitons, and Similaritons. NY: Institute of Optics, University of Rochester; 2008

[15] Zhou Q, Liu L, Zhang H, Wei C, Lu J, Yu H, et al. Analytical study of Thirring optical solitons with parabolic law nonlinearity and spatio-temporal dispersion. The European Physical Journal - Plus. 2015;**130**:138

[16] Jiang Q, Su Y, Nie H, Ma Z, Li Y. New type gray spatial solitons in twophoton photorefractive media with both the linear and quadratic electro-optic effects. Journal of Nonlinear Optical Physics & Materials. 2017;**26**(1): 1750006 (9 pp)

[17] Topkara E, Milovic D, Sarma AK, Zerrad E, Biswas A. Optical solitons with non-Kerr law nonlinearity and inter-modal dispersion with time-dependent coefficients. Communications in Nonlinear Science and Numerical Simulation. 2010;**15**: 2320-2330

[18] Jovanoski Z, Roland DR. Variational analysis of solitary waves in a homogeneous cubic-quintic nonlinear medium. Journal of Modern Optics. 2001;**48**:1179

*Bright, Dark, and Kink Solitary Waves in a Cubic-Quintic-Septic-Nonical Medium DOI: http://dx.doi.org/10.5772/intechopen.92819*

[19] Stegeman GI, Stolen RH. Waveguides and fibers for nonlinear optics. Journal of the Optical Society of America B: Optical Physics. 1989;**6**:652

**References**

[1] Hasegawa A, Tappert FD.

Physics Letters. 1973;**23**:142

[2] Hasegawa A, Tappert FD.

Physics Letters. 1973;**23**:171

Letters. 1980;**45**:1095

[3] Mollenauer LF, Stolen RH,

Soliton propagation and selfconfinement of laser-beams by Kerr

optical nonlinearity. Optics Communication. 1985;**55**:201

Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Applied

*Nonlinear Optics - From Solitons to Similaritons*

nonlinear-fiber amplifier with an external source. Physical Review A.

[12] Goyal A, Gupta R, Kumar CN, Raju TS. Chirped femtosecond solitons and double-kink solitons in the cubicquintic nonlinear Schrödinger equation with self-steepening and self-frequency shift. Physical Review A. 2011;**84**:

[13] Porubov AV, Andrievsky BR. Kink and solitary waves may propagate together. Physical Review E. 2012;**85**:

[14] Agrawal GP. Optical Solitons, Autosolitons, and Similaritons. NY: Institute of Optics, University of

[15] Zhou Q, Liu L, Zhang H, Wei C, Lu J, Yu H, et al. Analytical study of Thirring optical solitons with parabolic law nonlinearity and spatio-temporal dispersion. The European Physical Journal - Plus. 2015;**130**:138

[16] Jiang Q, Su Y, Nie H, Ma Z, Li Y. New type gray spatial solitons in twophoton photorefractive media with both the linear and quadratic electro-optic effects. Journal of Nonlinear Optical Physics & Materials. 2017;**26**(1):

[17] Topkara E, Milovic D, Sarma AK, Zerrad E, Biswas A. Optical solitons with non-Kerr law nonlinearity and inter-modal dispersion with time-dependent coefficients.

Communications in Nonlinear Science and Numerical Simulation. 2010;**15**:

[18] Jovanoski Z, Roland DR. Variational

homogeneous cubic-quintic nonlinear medium. Journal of Modern Optics.

analysis of solitary waves in a

2010;**81**:043820

063830

046604

Rochester; 2008

1750006 (9 pp)

2320-2330

2001;**48**:1179

Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion. Applied

Gordon JP. Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Physical Review

[4] Barthelemy A, Maneuf S, Froehly C.

[5] Hasegawa A, Kodama Y. Solitons in Optical Communications. Oxford: Oxford University Press; 1995

[6] Agrawal GP. Nonlinear Fiber Optics. New York: Academic Press; 2013

Khabibullaev P. Optical Solitons. Berlin:

[8] Kivshar YS, Luther-Davies B. Dark optical solitons: Physics and applications.

[9] Kivshar YS, Agrawal GP. Optical Solitons: "From Fibers to Photonic Crystal". San Diego: Academic Press;

[10] Agrawal GP, Headley C III. Kink solitons and optical shocks in dispersive nonlinear media. Physical Review A.

[11] Raju TS, Panigrahi PK. Self-similar

propagation in a graded-index

[7] Abdullaev F, Darmanyan S,

Physics Reports. 1998;**298**:81

Springer-Verlag; 1991

2003

**222**

1992;**46**:1573

[20] Serkin VN, Belyaeva TL, Alexandrov IV, Melchor GM. Optical pulse and beam propagation III. In: Band YB, editor. SPIE Proceedings. Vol. 4271. Bellingham: SPIE; 2001. p. 292

[21] Dai CQ, Wang YY, Zhang JF. Analytical spatiotemporal localizations for the generalized (3+1)-dimensional nonlinear Schrödinger equation. Optics Letters. 2010;**35**:1437

[22] Dai CQ, Zhang JF. Exact spatial similaritons and rogons in 2D gradedindex waveguides. Optics Letters. 2010; **35**:2651

[23] Dai CQ, Zhu SQ, Wang LL, Zhang JF. Exact spatial similaritons for the generalized (2+1)-dimensional nonlinear Schrödinger equation with distributed coefficients. Europhysics Letters. 2010;**92**:24005

[24] Dai CQ, Wang XG, Zhang JF. Nonautonomous spatiotemporal localized structures in the inhomogeneous optical fibers: interaction and control. Annals of Physics (NY). 2011;**326**:645

[25] Dai CQ, Yang Q, He JD, Wang YY. Nonlinear tunneling effect in the (2+1) dimensional cubic-quintic nonlinear Schrödinger equation with variable coefficients. The European Physical Journal. 2011;**D63**:141

[26] Jayabalan J, Singh A, Chari R, Khan S, Srivastava H, Oak SM. Transient absorption and higher-order nonlinearities in silver nanoplatelets. Applied Physics Letters. 2009;**94**: 181902

[27] Reyna AS, Jorge KC, de Araújo CB. Two-dimensional solitons in a

quintic-septimal medium. Physical Review A. 2014;**90**:063835

[28] Triki H, Porsezian K, Dinda PT, Grelu P. Dark spatial solitary waves in a cubic-quintic-septimal nonlinear medium. Physical Review A. 2017;**95**: 023837

[29] Triki H, Porsezian K, Choudhuri A, Tchofo Dinda P. Chirped solitary pulses for a nonic nonlinear Schrödinger equation on a continuous-wave background. Physical Review A. 2016; **93**:063810

[30] Boyd RW, Lukishova SG, Shen YR, editors. Self-focusing: "Past and present (fundamentals and prospects)". In: Topics in Applied Physics. Vol. 114. Berlin: Springer; 2009

[31] Zeng J, Malomed BA. Bright solitons in defocusing media with spatial modulation of the quintic nonlinearity. Physics Review. 2012;**E86**:036607

[32] Couairon A, Mysyrowicz A. Femtosecond filamentation in transparent media. Physics Reports. 2007;**441**:47

[33] Liu W, Petit S, Becker A, Akцozbek N, Bowden CM, Chin SL. Intensity clamping of a femtosecond laser pulse in condensed matter. Optics Communication. 2002;**202**:189

[34] Mohamadou A, Latchio Tiofack CG, Kofané TC. Wave train generation of solitons in systems with higher-order nonlinearities. Physical Review E. 2010; **82**:016601

[35] Shukla PK, Rasmussen JJ. Modulational instability of short pulses in long optical fibers. Optics Letters. 1986;**11**:171

[36] Potasek MJ. Modulation instability in an extended nonlinear Schrödinger equation. Optics Letters. 1987;**12**:921

**Chapter 11**

**Abstract**

**1. Introduction**

**225**

*and Alidou Mohamadou*

magnitude, when the walk-off increases.

delay response time, cross-phase modulation

Emergence of Raman Peaks

Due to Septic Nonlinearity

*Michel-Rostand Soumo Tchio, Saïdou Abdoulkary*

in Noninstantaneous Kerr Media

We analyze the modulation instability induced by cross-phase modulation of two co-propagating optical beams in nonlinear fiber with the effect of higher-order dispersion and septic nonlinearity. We investigate in detail the effect of relaxation nonlinear response to the gain spectrum both in normal group velocity dispersion

relaxation nonlinear response time as well as the higher-order process particularly influence the generation of the modulation instability gain. Our results shows that the emerging Raman peaks is observable both in the case of weak dispersion and in a higher-order dispersion for mixed GVD regime with slow response time. These Raman peaks are shifted toward higher frequencies with the decrease of their

(GVD) and anomalous dispersion regime. We show that the walk-off, the

**Keywords:** septic nonlinearity, higher-order dispersions, walk-off effects,

The generation of a wave train is a preoccupying subject in the realm of nonlinear science. This is mainly due to two effects: nonlinearity and dispersion. These two notions are essential in the propagation of the wave over long distances and the optical pulse resulting from this interaction gives rise to an optical soliton. The dynamic evolution of nonlinear pulses in nonlinear optical systems can be modeled by the well known nonlinear Schrödinger (NLS) equation which represents the lowest-order nontrivial condition describing the propagation process [1]. The co-propagation of two nonlinear waves in nonlinear optical Kerr media under a slowly varying amplitude approximation is made by using extensions of the NLS equation, whose analytical results provide the dispersion relation, the unstable conditions, as well as the gain spectra. This extension of NLS equation can take into account a large variety of physical properties such as higher-orders dispersion (like third-order dispersion (TOD) and fourth-order dispersion (FOD)) [2–9]; multiple optical beams [10]; negative index material [11]; saturable nonlinearity [12]; and non-instantaneous nonlinear response [13]. Third-order dispersion is used to describe the proprieties of ultrashort pulses in the subpicosecond to femtosecond domain. Usually in nonlinear optic, Kerr nonlinearity is used to compensate the

[37] Porsezian K, Nithyanandan K, Vasantha Jayakantha Raja R, Shukla PK. Modulational instability at the proximity of zero dispersion wavelength in the relaxing saturable nonlinear system. Journal of the Optical Society of America. 2012;**B29**:2803

[38] Reyna AS, Malomed BA, de Araújo CB. Stability conditions for onedimensional optical solitons in cubicquintic septimal media. Physical Review A. 2015;**92**:033810

[39] Palacios SL, Guinea A, Fernández-Díaz JM, Crespo RD. Dark solitary waves in the nonlinear Schrodinger equation with third order dispersion, self-steepening, and self-frequency shift. Physics Review. 1999;**E60**:R45

[40] Youssoufa M, Dafounansou O, Mohamadou A. W-shaped, dark and grey solitary waves in the nonlinear Schrödinger equation competing dual power-law nonlinear terms and potentials modulated in time and space. Journal of Modern Optics. 2019;**66**(5): 530-540

[41] Tang XY, Shukla PK. Solution of the one-dimensional spatially inhomogeneous cubic-quintic nonlinear Schrodinger equation with an external potential. Physical Review A. 2007;**76**: 013612

[42] Boyd RW, Raymer MG, Narducci LM, editors. Optical Instabilities. London: Cambridge University Press; 1986

[43] Arecchi FT, Harrison RG, editors. Instabilities and Chaos in Quantum Optics. Berlin: Springer-Verlag; 1987

[44] Weiss CO, Vilaseca R. Dynamics of Lasers. New York: Weinheim; 1991

[45] van Tartwijk GHM, Agrawal GP. Progress in Quantum Electronics. 1998; **22**:43

## **Chapter 11**

[37] Porsezian K, Nithyanandan K, Vasantha Jayakantha Raja R, Shukla PK.

proximity of zero dispersion wavelength in the relaxing saturable nonlinear system. Journal of the Optical Society of

*Nonlinear Optics - From Solitons to Similaritons*

Araújo CB. Stability conditions for onedimensional optical solitons in cubicquintic septimal media. Physical Review A. 2015;**92**:033810

[39] Palacios SL, Guinea A, Fernández-Díaz JM, Crespo RD. Dark solitary waves in the nonlinear Schrodinger equation with third order dispersion, self-steepening, and self-frequency shift. Physics Review. 1999;**E60**:R45

[40] Youssoufa M, Dafounansou O, Mohamadou A. W-shaped, dark and grey solitary waves in the nonlinear Schrödinger equation competing dual power-law nonlinear terms and

potentials modulated in time and space. Journal of Modern Optics. 2019;**66**(5):

[41] Tang XY, Shukla PK. Solution of the

inhomogeneous cubic-quintic nonlinear Schrodinger equation with an external potential. Physical Review A. 2007;**76**:

[43] Arecchi FT, Harrison RG, editors. Instabilities and Chaos in Quantum Optics. Berlin: Springer-Verlag; 1987

[44] Weiss CO, Vilaseca R. Dynamics of Lasers. New York: Weinheim; 1991

[45] van Tartwijk GHM, Agrawal GP. Progress in Quantum Electronics. 1998;

one-dimensional spatially

[42] Boyd RW, Raymer MG, Narducci LM, editors. Optical Instabilities. London: Cambridge

University Press; 1986

530-540

013612

**22**:43

**224**

Modulational instability at the

[38] Reyna AS, Malomed BA, de

America. 2012;**B29**:2803
