Complex behavior and soliton solutions of the Resonance Nonlinear Schrödinger equation with modified extended tanh expansion method and Galilean transformation

Abstract

This paper delves into a complex mathematical equation known as the resonance nonlinear Schrödinger equation. We analyze its detailed patterns and solutions, explaining the fundamental algorithm of the equation and simplifying it into an ordinary differential equation. Additionally, we use the Galilean transformation to turn it into a set of simpler equations. Our investigation covers various aspects such as bifurcations, chaotic flows, and other interesting dynamic features. This culminates in identifying and visually representing solitary wave solutions. We thoroughly examine and present cases ranging from an elegant solitary wave set against a repeating background with unique characteristics to periodic solitons and singular breather-like waves. This work represents a significant step forward in understanding the complex and unpredictable behavior of this mathematical model.