Bifurcation and Stochastic Dynamics of the Hirota-Maccari System: A Study of Noise-Induced Solitons

Abstract

This study aims to investigate the intricate dynamics of the stochastic Hirota-Maccari system forced in the It\(\hat{o}\) sense. First, we establish a dynamical system linked to the equation by employing the Galilean transformation. By employing the system of complete discriminant of the polynomial technique, we methodically develop single traveling wave solutions for the governing model. Our solutions encompass hyperbolic, rational, and trigonometric forms, Jacobian elliptic functions, and various solitary wave solutions, along with transitions of Jacobian elliptic functions to periodic and hyperbolic solutions. Furthermore, we investigate the bifurcation processes that are intrinsic to the derived system using concepts from the theory of planar dynamical systems. Additionally, the existence of chaotic behaviors in the governing model is investigated by adding a perturbed term into the resulting dynamical system and presenting various two and three dimensional phase pictures. We also conduct sensitivity analyses to understand how various initial conditions affect the governing model. The proposed bifurcation and sensitivity analyses provide a framework for predicting and managing soliton behaviour in noisy environments, with possible applications in optical communications, fluid dynamics, and quantum mechanics. To illustrate our findings, we include several graphics that vividly demonstrate the influence of noise. These graphics reveal distinct patterns of random fluctuations, demonstrating the tremendous impact of stochastic forces across different systems and scenarios.