1. Introduction

The numerical simulation and analytical models of nonlinear Schrödinger equation (NLSE) play important roles in the design optimization of optical communication systems. They help to understand the underlying physics phenomena of the ultrashort pulses in the nonlinear and dispersion medium.

The inverse scattering [1], variation, and perturbation methods [2] could obtain the analytical solutions under some special conditions. These included the inverse scattering method for classical solitons [3], the dam-break approximation for the non-return-to-zero pulses with the extremely small chromatic dispersion [4], and the perturbation theory for the multidimensional NLSE in the field of molecular physics [5]. When a large nonlinear phase was accumulated, the Volterra series approach was adopted [6]. With the assumption of the perturbations, the NLSE with varying dispersion, nonlinearity, and gain or absorption parameters was solved in [7]. In [8], the generalized Kantorovitch method was introduced in the extended NLSE. By introducing Rayleigh's dissipation function in Euler-Lagrange equation,

the algebraic modification projected the extended NLSE as a frictional problem and successfully solved the soliton transmission problems [9].

Since the numerical computation of solving NLSE is a huge time-consuming process, the fast algorithms and efficient implementations, focusing on (i) an accurate numerical integration scheme and (ii) an intelligent control of the longitudinal spatial step size, are required.

The finite differential method [10] and the pseudo-spectral method [11] were adopted to increase accuracy and efficiency and suppress numerically induced spurious effects. The adaptive spatial step size-controlling method [12] and the predictor-corrector method [13] were proposed to speed up the implementation of split-step Fourier method (SSFM). The cubic (or higher order) B-splines were used to handle nonuniformly sampled optical pulse profiles in the time domain [14]. The Runge-Kutta method in the interaction picture was applied to calculate the effective refractive index, effective area, dispersion, and nonlinear coefficients [15].

<sup>ϕ</sup>ð Þ <sup>z</sup>;<sup>ω</sup> is caused by the nonlinear effect, and <sup>ϕ</sup>ð Þ¼ <sup>z</sup>;<sup>ω</sup> <sup>Ð</sup> <sup>z</sup>

�adz=<sup>2</sup> exp ð Þ <sup>β</sup>1ωdz exp <sup>β</sup>2=2ω<sup>2</sup>

P zð Þ <sup>þ</sup> dz;<sup>ω</sup> <sup>p</sup> exp ½ � <sup>i</sup>φð Þ <sup>z</sup> <sup>þ</sup> dz:<sup>ω</sup>

P zð Þ ;<sup>ω</sup> <sup>p</sup> can be regarded as

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi P zð Þ ;<sup>ω</sup> <sup>p</sup> <sup>≈</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

> �adz=2 �

Re h i P zð Þ exp <sup>i</sup>ωβ1dz <sup>þ</sup> <sup>i</sup>ω<sup>2</sup>

dz � �ΔP zð Þþ ;<sup>ω</sup> sin <sup>1</sup>

<sup>β</sup>2dz � � <sup>¼</sup> cos ωβ1dz <sup>þ</sup> <sup>ω</sup><sup>2</sup>

h i P zð Þ is the average signal intensity. ΔP zð Þ ;ω is the noise or modulation term.

Schematic illustration of medium. u(z, t) and u(z + dz, t) correspond to the field amplitudes at z and z + dz,

h i P zð Þ <sup>p</sup> <sup>1</sup> <sup>þ</sup>

The operation exp <sup>i</sup>ωβ1dz <sup>þ</sup> <sup>i</sup>ω<sup>2</sup> ð Þ <sup>β</sup>2dz can be split into its real and imaginary

The modulation or noise ΔP zð Þ þ dz;ω is ΔP zð Þ þ dz;ω ≈P zð Þ� þ dz;ω h i P zð Þ

2 β2ω<sup>2</sup>

� � (8)

<sup>β</sup>2dz � � <sup>Δ</sup>P zð Þ ;<sup>ω</sup>

� � � �

<sup>β</sup>2dz � � <sup>þ</sup> <sup>i</sup>sin ωβ1dz <sup>þ</sup> <sup>ω</sup><sup>2</sup>

dz � �2h i P zð Þ <sup>φ</sup>ð<sup>z</sup> <sup>þ</sup> dz;ω<sup>Þ</sup>

The small-signal theory implies that the frequency modulation or noise

dt is small enough. Finally ([21])

P zð Þ ;<sup>ω</sup> <sup>p</sup> exp <sup>i</sup>φð Þþ <sup>z</sup>;<sup>ω</sup> <sup>i</sup><sup>γ</sup> <sup>P</sup> <sup>þ</sup> <sup>2</sup>P<sup>0</sup> f g ½ �dz

dz � � ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ΔP zð Þ ;ω

P zð Þ ;<sup>ω</sup> <sup>p</sup> <sup>e</sup>

<sup>2</sup>P zð Þ � � (5)

<sup>2</sup>h i P zð Þ <sup>þ</sup> <sup>i</sup>φð Þ <sup>z</sup> <sup>þ</sup> dz;<sup>ω</sup>

β2dz � � (7)

u zð Þ¼ <sup>þ</sup> dz;<sup>ω</sup> exp dzD^ � � ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Assuming: P zð Þ¼ ;ω h i P zð Þ þ ΔP zð Þ ;ω

<sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

¼ e

[3].

Figure 1.

respectively.

u zð Þ þ dz;ω (is)

Nonlinear Schrödinger Equation

DOI: http://dx.doi.org/10.5772/intechopen.81093

There is [20] h i P zð Þ ΔP zð Þ ;ω The amplitude ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>φ</sup>\_ð Þ¼ <sup>z</sup> <sup>þ</sup> dz;<sup>ω</sup> <sup>d</sup>φ\_ð Þ <sup>z</sup>þdz;<sup>ω</sup>

exp <sup>i</sup>ωβ1dz <sup>þ</sup> <sup>i</sup>ω<sup>2</sup>

P zð Þ¼ <sup>þ</sup> dz; <sup>ω</sup> <sup>e</sup>�adz=2�iωβ1dz

cos 1 2 β2ω<sup>2</sup>

parts:

So

And

17

P zð Þ¼ þ dz;ω h i P zð Þ þ 2e

<sup>0</sup> γ P zð Þþ ;ω 2P<sup>0</sup> ½ � ð Þ z;ω dz

iφð Þ zþdz;ω

(4)

(6)

Recently, the generalized NLSE, taking into account the dispersion of the transverse field distribution, is derived [16]. By an inhomogeneous quasi-linear firstorder hyperbolic system, the accurate simulations of the intensity and phase for the Schrödinger-type pulse propagation were obtained [17]. It has been demonstrated that modulation instability (MI) can exist in the normal GVD regime in the higherorder NLSE in the presence of non-Kerr quintic nonlinearities [18].

In this chapter, several methods to solve the NLSE will be presented: (1) The small-signal analysis theory and split-step Fourier method to solve the coupled NLSE problem, the MI intensity fluctuation caused by SPM and XPM, can be derived. Furthermore, this procedure is also adapted to NLSE with high-order dispersion terms. The impacts of fiber loss on MI gain spectrum can be discussed. The initial stage of MI can be described, and then the whole evolution of MI can also be discussed in this way; (2) the Green function to solve NLSE in the time domain. By this solution, the second-, third-, and fourth-order dispersion coefficients is discussed; and (3) the traveling wave solution to solve NLSE for ASE noise and its probability density function.
