Bifurcation Analysis, Modulational Instability and Solitons Solutions for a Coupled Nonlinear Schr\"odinger-Type Equations with Variable Coefficients

TL;DR

This paper investigates the complex dynamics of a coupled nonlinear Schrödinger system with variable coefficients, revealing bifurcations, soliton solutions, and modulational instability, with implications for understanding nonlinear wave phenomena.

Contribution

It provides a novel analysis of bifurcation, stability, and instability in a coupled variable-coefficient nonlinear Schrödinger system, including the emergence of solitons and chaotic behavior.

Findings

Existence of Jacobi elliptic and soliton solutions

System exhibits chaotic behavior under perturbations

Modulational instability occurs under certain parameter conditions

Abstract

Understanding, predicting, and controlling physical processes often relies on the analysis of the dynamics of partial differential equations (PDEs). In this context, the present study offers an in-depth investigation into the nonlinear dynamics of a novel coupled nonlinear Schr\"odinger system with variable coefficients. To begin with, the dynamics of traveling-wave solutions is analyzed through bifurcation theory, which uncovers the presence of Jacobi elliptic functions and solitonic structures. Then, a sensitivity analysis is carried out to confirm the numerical relevance and stability of these solutions. However, when subjected to external perturbations, the system exhibits a tendency toward chaotic behavior. Finally, the results point to the emergence of modulational instability under specific configurations of the dispersion and nonlinearity parameters.