Robustness of the solitons against perturbations in certain nonlocal nonlinear Schr\"{o}dinger type equations in Nonlinear Physics

TL;DR

This paper investigates the stability of soliton solutions in four types of nonlocal nonlinear Schrödinger equations, demonstrating their robustness under perturbations across various physical contexts like optics and Bose-Einstein condensates.

Contribution

It introduces a novel stability analysis approach for solitons in nonlocal nonlinear equations, confirming their robustness in multiple physical models.

Findings

Soliton solutions are stable under perturbations in all four equations.

The stability analysis method is new and applicable to various nonlocal nonlinear systems.

Results support the physical relevance of these solutions in real-world scenarios.

Abstract

The nonlocal nonlinear evolution equations describe phenomena in which wave evolution is influenced by local and nonlocal spatial and temporal variables. These equations have opened up a new wave of physically important nonlinear evolution equations. Their solutions provide insights into the interplay between nonlinearity and nonlocality, making it a cornerstone in the study of nonlocal nonlinear systems. However, the stability of such solutions has not been extensively explored in the literature. Stability analysis ensures that these solutions are robust and capable of persisting under real-world perturbations, making them physically meaningful. In this work, we examine the stability of soliton solutions of four types of nonlocal nonlinear evolutionary equations: (i) the space-shifted nonlocal nonlinear Schr\"{o}dinger equation, (ii) the nonlocal complex time-reversed Hirota equation, (iii) the nonlocal real space-time-reversed modified Korteweg-de Vries equation, and (iv) a fourth-order nonlocal nonlinear Schr\"{o}dinger equation. These equations arise in various physical fields such as nonlinear optics, Bose-Einstein condensates, plasmas and so on where nonlocality and nonlinearity play significant roles. We introduce certain perturbations to the soliton solutions of these equations and analyze their stability. Our findings indicate that the soliton solutions of the aforementioned equations are stable under such perturbations. To the best of our knowledge, this approach to investigating the stability of these solutions is novel.