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

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. •Explore intricate dynamics, unveil solitary wave solutions in Resonance Nonlinear Schrödinger Equation (RNLSE).•Elucidate RNLSE’s algorithm, transforming it into an ODE for enhanced analytical insight.•Use Galilean transformation to convert RNLSE into a system of ODEs, broadening dynamical exploration.•Thoroughly examine, and visually present diverse solitary wave solutions, advancing comprehension of RNLSE’s complex behavior.