Exploring new solutions for the stochastic Gardner equation forced by advection Brownian motion used in nonlinear wave propagation

Abstract

Background

The stochastic Gardner equation with Itô-type advection Brownian motion provides an effective mathematical framework for modeling the propagation of nonlinear waves in random environments. It plays a key role in understanding the transition from coherent wave structures to irregular dynamics by elucidating the interplay among randomness, nonlinearity, and dispersion in complex physical systems. Consequently, obtaining exact analytical solutions of the stochastic Gardner equation is of significant theoretical and practical importance.

Methods

By employing appropriate transformation techniques together with Itô calculus, the stochastic Gardner equation is decomposed into two coupled components: a deterministic Gardner equation with an additional diffusion term and a stochastic ordinary differential equation. The extended tanh-function method is applied to derive exact traveling wave solutions of the deterministic Gardner equation. These solutions are then combined with the analytical solution of the stochastic ordinary differential equation to construct exact solutions of the original stochastic Gardner equation. The effectiveness of the proposed framework can be demonstrated by deriving various stochastic wave solutions and graphing them, thus proving its ability to investigate nonlinear wave propagation in the presence of stochastic effects.

Results

A variety of exact analytical solutions for the stochastic Gardner equation are successfully obtained, including solitary wave structures influenced by stochastic effects. The impact of advection Brownian motion on the wave dynamics is systematically investigated. Three-dimensional graphical simulations, generated using MATLAB, illustrate how stochastic advection modifies the shape, amplitude, and evolution of the solutions compared to their deterministic counterparts. Additionally, these findings shed light on how stochastic perturbations affect the amplitude and propagation characteristics of nonlinear waves in the Gardner model.

Conclusion

The proposed analytical framework provides explicit exact solutions for the stochastic Gardner equation and reveals the significant role of advection Brownian motion in altering nonlinear wave behavior. These results enhance the understanding of stochastic nonlinear wave propagation and may be useful for modeling realistic physical systems subject to random perturbations.