Novel Solitary Wave Solutions and Conservation Laws of the Stochastic Biswas–Milovic Equation

Abstract

This study presents novel analytical solutions for the stochastic Biswas-Milovic equation (SBME) with dual-power law nonlinearity, a critical model for wave propagation in noisy nonlinear systems. For the first time, we derive bright, dark, and singular soliton solutions under multiplicative noise through detailed three-dimensional visualizations, showcasing their structural features and dynamic behaviors using an innovative hybrid approach combining the \(\phi ^{6}-\) model expansion and extended simplest equation methods. We develop a rigorous mathematical framework to derive these solutions. Moreover, novel conservation laws associated with the SBME are derived, highlighting the conservative properties and their significance in broader scientific and engineering applications. These findings contribute new insights into the study of stochastic nonlinear systems and expand the scope of soliton theory. The SBME is crucial for numerous engineering applications, particularly in systems characterized by randomness, such as diffusion and Brownian motion. The influence of multiplicative noise on soliton propagation is analyzed, revealing conditions under which these solitons persist despite stochastic perturbations.