On the Analytical and Numerical Solutions of the Linear Damped NLSE for Modeling Dissipative Freak Waves and Breathers in Nonlinear and Dispersive Mediums: An Application to a Pair-Ion Plasma - El-Tantawy, S. A.; Salas, Alvaro H.; Alharthi, M. R. (Frontiers in Physics, 2021)
In this work, two approaches have been introduced for solving a linear damped nonlinear Schrödinger equation (NLSE) for modelling the dissipative rogue waves (DRWs) and dissipative breathers (DBs). It is known that the linear damped NLSE is considered a non-integrable differential equation. Thus, it does not support an explicit analytic solution till now due to the presence of the linear damping term. Consequently, two accurate solutions will be derived and obtained in details. The first solution is called a semi-analytical solution while the second one is 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 initial solution for solving the linear damped NLSE. With respect to the approximate numerical solution, the moving boundary method (MBM) with the help of finite differences method (FDM) will be devoted for this purpose. The maximum residual (local and global) errors formula for the semi-analytical solution will be derived and obtained. Also, the numerical values of both the 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 via using the nonlinear polynomial based on 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 help us to interpret and understand the dynamic behavior of modulated structures in various plasma models, fluid mechanics, optical fiber, Bose-Einstein condensate, etc.