Dynamics of stochastic nonlinear waves in fractional complex media

Abstract

We consider in this work a nonlinear heat equation with cubic nonlinearity (alias real-valued stochastic Ginzburg–Landau equation) with spatiotemporal variable fractional derivatives, which is forced by a multiplicative noise in the Itô sense. Using an appropriate transformation, the model equation is reduced into a second-order nonlinear ordinary differential equation. Applying the method of Riccati equation and combining the ${\varphi }^4$ expansion method with the method of Weierstrass elliptic function, exact analytical solutions of various types, including cnoidal wave solutions, solitonlike wave solutions, as well as symmetry wave solutions are presented. We show how exact analytical solutions may contribute to a deeper understanding of nonlinear behaviors, spatiotemporal correlations, and stochastic fluctuations in complex media. More precisely, we employ exact solutions to investigate graphically the effects of both fractionality and multiplicative noise on the stochastic nonlinear wave in complex fractional media whose dynamics are described by the equation model under consideration. One of the novelty of our work is that the symmetry exact wave solutions found here have not yet been presented in the context of stochastic nonlinear waves. Our results show that the variable order fractional derivatives can be used for controlling the dynamical properties of the complex systems governed by the model equation under consideration.