Analytical Approaches to the Stochastic Nonlinear Kodama Equation via the Impact of Multiplicative Noise

ABSTRACT This study focuses on the stochastic nonlinear Kodama (SNLK) equation influenced by multiplicative noise interpreted in the Stratonovich sense. The equation is significant for modeling complex physical systems in fluid dynamics, plasma physics, and nonlinear optics. We employ both the Kumar–Malik method and the polynomial expansion technique to construct exact soliton solutions in various functional forms, including Jacobi elliptic, hyperbolic, exponential, and trigonometric types. The combination of these two analytical methods provides a novel framework for investigating nonlinear wave structures in stochastic environments. To complement the analytical results, numerical simulations are conducted using Maple (or Mathematica) software, which confirm the accuracy and dynamic consistency of the solutions. These simulations also offer deeper insight into the behavior and evolution of the solutions under random influences. Our results demonstrate the effectiveness and adaptability of the applied methods, making them suitable for a wider class of nonlinear partial differential equations. This work contributes to a better theoretical and computational understanding of stochastic wave phenomena and lays the groundwork for further exploration in related physical contexts.