New traveling wave solutions, phase portrait and chaotic pattern for the stochastic modified Korteweg–de Vries equation

Abstract

This article mainly studies the new traveling wave solutions of the stochastic modified Korteweg–de Vries equation with multiplicative noise. The traveling wave solutions in the form of hyperbolic function, trigonometric function, rational function and Jacobi elliptic function are obtained by transforming the modified Korteweg–de Vries equation into an ordinary differential equation through traveling wave transformation, and combining it with the complete discriminant system of polynomials. This article fully applies visual analysis technology and provides graphical representations of the obtained solutions in multiple dimensions such as phase maps, sensitivity maps, 3D maps and 2D maps, providing an intuitive and convenient channel for people to further understand the physical characteristics of the solutions for the stochastic modified Korteweg–de Vries equation. This article provides a rich graphical analysis of the solutions to the stochastic modified Korteweg–de Vries equation from multiple perspectives, and provides a fast and efficient solution method for the exact solution of the equation. The research method in this paper can also be used to study nonlinear partial differential equation in other fields.