Novel solitary wave solutions for stochastic nonlinear reaction–diffusion equation with multiplicative noise

Abstract The primary objective of this paper is to explore solitary wave solutions within the context of the stochastic Reaction–Diffusion equation, a problem that has not been previously investigated with respect to multiplicative noise. This marks a novel aspect of the current study. To accomplish this goal, we utilize two distinct integration methods: the extended Kudryashov’s method and the $$\left( \frac{G'}{G}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mfrac> <mml:msup> <mml:mi>G</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mi>G</mml:mi> </mml:mfrac> </mml:mfenced> </mml:math> -expansion method. These methods are chosen for their efficacy in handling nonlinear differential equations and are applied in tandem to uncover new trigonometric and hyperbolic stochastic exact solutions. The employment of these methods is crucial in generating solutions that have not been previously identified in this particular context. Subsequently, to assess the validity and usefulness of these newly discovered solutions, we conduct numerical simulations. These simulations involve implementing the derived solutions across a variety of scenarios or samples, allowing for an evaluation of their performance and applicability. This paper represents a significant contribution to the field of stochastic Reaction–Diffusion equations by introducing a pioneering investigation into the effects of multiplicative noise on solitary wave solutions. Furthermore, it offers a fresh perspective on obtaining exact solutions in the presence of such noise, showcasing the potential of the integration methods in tackling complex nonlinear stochastic systems.