Abstract
In this work, two approaches are introduced to solve a linear damped nonlinear Schrödinger equation (NLSE) for modeling the dissipative rogue waves (DRWs) and dissipative breathers (DBs). The linear damped NLSE is considered a non-integrable differential equation. Thus, it does not support an explicit analytic solution until now, due to the presence of the linear damping term. Consequently, two accurate solutions will be derived and obtained in detail. The first solution is called a semi-analytical solution while the second is an approximate numerical solution. In the two solutions, the analytical solution of the standard NLSE (i.e., in the absence of the damping term) will be used as the initial solution to solve the linear damped NLSE. With respect to the approximate numerical solution, the moving boundary method (MBM) with the help of the finite differences method (FDM) will be devoted to achieve this purpose. The maximum residual (local and global) errors formula for the semi-analytical solution will be derived and obtained. The numerical values of both maximum residual local and global errors of the semi-analytical solution will be estimated using some physical data. Moreover, the error functions related to the local and global errors of the semi-analytical solution will be evaluated using the nonlinear polynomial based on the Chebyshev approximation technique. Furthermore, a comparison between the approximate analytical and numerical solutions will be carried out to check the accuracy of the two solutions. As a realistic application to some physical results; the obtained solutions will be used to investigate the characteristics of the dissipative rogue waves (DRWs) and dissipative breathers (DBs) in a collisional unmagnetized pair-ion plasma. Finally, this study helps us to interpret and understand the dynamic behavior of modulated structures in various plasma models, fluid mechanics, optical fiber, Bose-Einstein condensate, etc.
Highlights
The first one is called the semi-analytical method which was built depending on the exact analytical solution of the standard nonlinear Schrödinger equation (NLSE)
The semi-analytical solution of the linear damped NLSE is considered to be the first attempt at modeling the dissipative rogue waves (DRWs) and dissipative breathers (DBs) in plasmas or in any other physical medium like optical fiber and so on
The exact solution of the standard NLSE is used as the initial condition to solve the linear damped NLSE using the hybrid moving boundary method (MBM)-finite difference method (FDM)
Summary
In the past few years, interest has increased in the treatment required to solve differential equations analytically and numerically which led to the interpretation of ambiguous nonlinear phenomena that exist and propagate in various fields of science such as the field of optical fiber, BoseEinstein condensate, biophysics, Ocean, physics of plasmas, etc. [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]. Few attempts have been made to understand the physical mechanism of generating and propagating DRWs and DBs (dissipative Akhmediev breathers (DABs) and dissipative Kuznetsov-Ma breathers (DKMBs) in the fluid mechanics and in plasma physics when considering the friction forces [45,46,47,48,49,50] In these papers, authors tried to find an approximate analytical solution to the following non-integrable linear damped NLSE.
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