## Abstract

We consider the nonlinear Schrödinger equation modified by a rational nonlinear term. The model appears in various studies often in the context of the Ginzburg-Landau equation. We investigate modulational instability by means of a linear stability analysis and show how the nonlinear terms affect the growth rate. This analytical result is confirmed by a numerical simulation. The latter analysis shows that breather-like solitons are generated from the instability, and the effects of the nonlinear terms are again clearly seen. Moreover, by employing an auxiliary-equation method we obtain kink and anti-kink soliton as analytical solutions. Our theoretical solution is in good agreement with our numerical investigation.

### Keywords

- generalized nonlinear schrödinger
- modulational instability
- breather like-soliton

## 1. Introduction

The nonlinear Schrödinger equation (NLSE) is the main equation which governs the propagation of pulses in various fields such as nonlinear optical systems, plasmas, fluid dynamics, Bose-Einstein condensation, and condensed matter physics. It has been shown to govern the evolution of a wave packets in weakly nonlinear and dispersive media and has thus arisen in such diverse fields. One other application of this equation is in pattern formation, where it has been used to model some nonequilibrium pattern forming systems. Most notable is the role it plays in understanding the propagation of electromagnetic waves in glass fibers and other optical waveguides [1].

In this paper we consider a NLS equation with inverse nonlinear terms. Inverse nonlinear term has been introduced for the first time by Malomed and al. [2] which has been later studied by [3, 4] in the case of the Ginzburg-Landau equation.

where

Modulational instability is one of the nonlinear wave equations associated to NLSE and appears in many physical systems. It indicates that due to the competition between nonlinearity and the dispersive effects, a small perturbation of the initial plane wave may induce an exponential growth of the wave amplitude, resulting in the carrier-wave breakup into a train of localized waves [6].

The NLSE is also one of the original nonlinear partial differential equations, the study of which has lead to fundamental advances in theoretical physics. The study of NLS was motivated by a large number of theoretical problems ranging from optical pulse propagation in nonlinear fibers to hydrodynamics, condensed matter physics and biophysics. It is now known that NLS is one of the few examples of completely integrable nonlinear partial differential equations [7, 8].

The main objective of this paper is to study MI in a generalized NLSE that includes rational nonlinear terms given by Eq. (1). By means of the linear stability analysis we explicitly investigate the stability condition of a launched plane wave. The results show that the MI gain is strongly dependent on the nonlinear parameters as well as the GVD. Our numerical simulations show that those parameters contribute to the formation and the propagation of the soliton like-breather in the systems. Those parameters can generate either stable or unstable solitons. We also investigate analytical soliton solutions. By employing auxiliary equation method we obtain kink and antikink solutions of Eq. (1).

The rest of the paper is organized as follows. The model is introduced in Section 2, which is followed by the analysis of the MI of the CW solutions in Section 3, direct simulations shown the formation of modulated wave as well as breather like-solitons and their stability in Section 4. Exact analytical kink and antikink soliton solutions are reported in Section 5, and the paper is concluded by Section 6.

## 2. Modulational instability

The nonlinear Schrodinger Equation Eq. (1) has the simplest solution in the form of a continuous wave as

where

Substituting Eq. (2) into the NLS equation Eq. (1) and linearizing the resulting equations, we obtain a linear equation of

Looking for solutions to this function in the form of plane waves

where the wave number

with

The modulation instability gain is related to the imaginary part of

Figure 1 shows the instability gain as a function of the perturbation wave number

## 3. The numerical simulation analysis

Analytical analysis done by linear stability shown the possibility of the formation of modulated waves in the consider system. This prediction can be numerically confirmed. In this way let us launch as initial condition a modulated plane wave:

From Figure 4 one can see the formation of bright solitary wave. The left hand top panel shows the generation of a pulse train toward the boundary regions but the intensity is smallest at the center. On right hand panel we can see the bright solitary wave behaves like a breather soliton is forming. This may be a multisoliton quasiperiodic solutions. It can be seen that the breather solutions keep their oscillating shapes, while the wave packets move as periodic solitons along the

There is more breathers when

## 4. Exact analytical solutions to the consider stationary model

We now discuss about the analytical solution to the stationary NLS of Eq. (1). Suppose that

is the solution of Eq. (1) where

For solving this equation we set

This is a nonlinear ordinary differential equation which can be solve by the auxiliary equation method.

### 4.1 The auxiliary equation method

The auxiliary equation method has been defined by [9, 10] while it allows to find more and new multiple solutions for nonlinear partial differential equations. The main steps of using this method is summarized as follows.

For solving equation

we set

We seek for the solutions of Eq. (11) in the following generalized form

in which

where

The solution of Eq. (9), balancing

where

where

Having obtained exact solutions of the stationary NLSE Eq. (9), we will use them together to construct soliton solutions of the NLSE Eq. (1). In this case, the kink-soliton and anti-kink solutions of Eq. (1) can be written in the form

where

## 5. Conclusion

In the present study a generalized nonlinear Schrödinger equation with particular nonlinearities has been introduced. The model including rational nonlinearity that arise from Malomed model and describes the propagation of nonlinear surface waves on a plasma with a sharp boundary. We explicitly investigated MI gain by means of linear stability analysis. Results reveal that the nonlinear parameters are strongly influences the dynamics of the launched plane wave. We further tested the evolutionary modulate plan wave numerically, which indicates that those parameters allow the formation of breather-like soliton in the system as well as bright soliton. We have investigated analytical kink and anti-kink soliton too.

It would be particularly worthwhile to extend this study to the generalized NLS with time and space modulated nonlinearities and potentials. This could allow more stability and more formation of the breather-like soliton as well as the Akhmediev breather [11], Peregrine rogue wave [12], and Kuznetsov-Ma breather [13, 14] and even high-order rogue waves [15]. MI gain distributions could bring different nonlinear excitation pattern dynamics.

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