The Hybrid Finite Difference and Moving Boundary Methods for Solving a Linear Damped Nonlinear Schrödinger Equation to Model Rogue Waves and Breathers in Plasma Physics

Alvaro H Salas,S A El-Tantawy,Jairo E Castillo H,Luigi Rodino

doi:10.1155/2020/6874870

Abstract

In this paper, the dissipative and nondissipative modulated pulses such as rogue waves (RWs) and breathers (Kuznetsov-Ma breathers and Akhmediev breathers) that can exist and propagate in several fields of sciences, for example, plasma physics, have been analyzed numerically. For this purpose, the fluid dusty plasma equations with taking the kinematic dust viscosity into account are reduced to the linear damped nonlinear Schrödinger equation using a reductive perturbation technique. It is known that this equation is not integrable and, accordingly, does not have analytical solution. Thus, for modelling both dissipative RWs and breathers, the improved finite difference method is introduced for this purpose. It is found that FDM is a good numerical technique for small time interval but for large time interval it becomes sometimes unacceptable. Therefore, to describe these waves accurately, the new improved numerical method is considered, which is called the hybrid finite difference method and moving boundary method (FDM-MBM). This last and updated method gives an accurate and excellent description to many physical results, as it was applied to the dust plasma results and the results were good.

Highlights

A nonlinear Schrodinger-type equation [1], especially the cubic nonlinearSchrodinger equation (CNLSE)(iztψ + Pz2xψ + Q|ψ|2ψ 0), is one of the most universal models that describe many physical nonlinear systems
It is understood that the values P and Q are functions in relevant physical configurational parameters such as temperature, pressure, particles density, entropy, and many other physical parameters according to the physical model in question. e cubic nonlinearSchrodinger equation (CNLSE) and many others differential equations were used to interpret many mysterious phenomena that occur in various fields of sciences [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. e CNLSE can support various exact analytic solutions; some of them depend on a zero background such as modulated envelope bright, dark, and gray-solitons [8, 9]
Several monographs and numerous published papers have been devoted to analyze the CNLSE, which possess special solution in the form of dark solitons which retain their velocities and shapes after interaction amongst themselves [18,19,20]. Some of these solutions are based on a finite background such as the breather structures (which include periodic space and localized time Akhmediev breathers (ABs) in addition to periodic time and localized space Kuznetsov-Ma (KM) soliton) and rogue waves (RWs) [10,11,12,13,14,15,16,17]

Summary

Introduction

(iztψ + Pz2xψ + Q|ψ|2ψ 0), is one of the most universal models that describe many physical nonlinear systems. E CNLSE is a good mathematical model for explaining a huge number of mysterious phenomena that appear in nature, mechanical systems, superfluidity in the absence of frictional forces such as viscosity, collision between charged and neutral particles, and so forth. E dCNLSE was derived for several plasma systems using the derivative expansion method in order to study its modulational instability (MI) of dissipative modulated structures including the dissipative/damping breathers and RWs in a collisional plasma [41,42,43,44,45,46]. Most of the previous studies focused on transforming the nonintegrable dCNLSE into the integrable CNLSE using an appropriate transformation [40] in order to investigate envelope solitons, breathers, RWs, cnoidal waves, and so forth. We will discuss a series of various solutions that are supported by the CNLSE

Physical Problem and Mathematical Model

Methods of Solution

FDM-MBM for Analyzing RWs and Breathers