Exact solutions of nonlinear stochastic Newell-Whitehead-Segel equation by a reduction technique

The Newell-Whitehead-Segel (NWS) model is a reaction-diffusion system that has been widely used to study pattern formation in biological and physical systems. In this paper, we present a powerful method for obtaining exact solutions of the stochastic NWS model by applying Nucci's reduction method. The method involves transforming the original stochastic NWS equation into a second-order nonlinear ordinary differential equation (ODE). Then, it can be solved exactly using the reduction of corresponding dynamical system into a first order ODE. We apply the method to the stochastic NWS model, including different nonlinear terms, and demonstrate the utility of the method in revealing new insights into the behavior of the system in the presence of noise and randomness. The Nucci reduction method, similar to other analytical techniques for solving differential equations, possesses both benefits and drawbacks. One significant advantage is its ability to yield diverse solution types, including soliton, hyperbolic, and wave solutions, among others, without imposing limitations on the attainable solutions. Furthermore, in specific instances, the method can also extract the first integral. The exact solutions obtained using the method provide a useful benchmark for comparing with numerical simulations and experimental data and can help guide the design of new materials and systems with desired pattern formation properties.