Bifurcation, Chaotic Behavior and Effects of Noise on the Solitons for the Stochastic Jaulent-Miodek Hierarchy Model

This study presents the dynamical analysis of the stochastic Jaulent-Miodek Hierarchy (SJMH) system under the effect of noise. The energy-dependent Schrödinger potential that is included in the SJMH equation is utilized in engineering systems, optics, condensed matter physics, and fluid dynamics. Therefore, it is essential to look into this dynamic problem from a mathematical perspective while considering the influence of Brownian motion. The system undergoes a certain transformation to become a planer dynamical system, and the bifurcation is analyzed Additionally, the sensitivity visualized is observed by introducing certain periodic pressures into the model under consideration, the quasi-periodic solution for the perturbed system is numerically studied. Two-dimensional phase pictures are shown concerning the parameter of the perturbed model. The Sardar subequation approach is used to explore the closed-form invariant solution known as solitons for the stochastic Jaulent-Miodek model. The different forms of soliton solitons are constructed in the form of bright, dark, dark-bright, periodic, and another form under the noise. It is noticed that the model has periodic oscillating nonlinear waves, various soliton profiles, and kink wave profiles. Using Wolfram Mathematica, some of the newly created soliton solutions are verified by reintegrating them into the relevant system for soft computation. The different effects of noise are plotted in the form of 3D, and 2D, and their corresponding contours by choosing suitable values of parameters.