Study on a Stochastic Riemann Wave Equation: Transformations, Multi-Wave Solutions and Auto Bäcklund Transformations

As is well known, transformations play a crucial role in studying nonlinear evolution equations—especially when constructing their analytical solutions. On one hand, transformations can simplify these equations; solving the simplified versions then allows us to obtain the solutions of the original equations. On the other hand, establishing transformations between different equations (or between different solutions of the same equation) enables further derivation of a series of new solutions for the original equations. This paper studies a stochastic Riemann wave equation. Specifically, new transformations are introduced to simplify the original equation into a couple of integer-order nonlinear evolution equations, followed by constructing various types of multi-wave mixed solutions or nonlinear superposition solutions via several efficient algorithms. Additionally, by establishing new auto Bäcklund transformations for the equation, a series of distinct new solutions can be further derived based on the previously obtained ones. These results provide important insights into stochastic wave phenomena and offer analytical tools for understanding wave behavior in physical contexts such as nonlinear optics and quantum mechanics, while advancing methods for stochastic nonlinear differential equations.