On Explicit Solutions for Coupled Reaction-Diffusion and Burgers-Type Equations with Variable Coefficients Through a Riccati System

TL;DR

This paper derives explicit solutions for complex coupled reaction-diffusion and Burgers-type equations with variable coefficients using similarity transformations and Riccati systems, revealing diverse dynamic behaviors.

Contribution

It introduces a method to obtain explicit solutions for variable coefficient nonlinear systems via Riccati systems and similarity transformations.

Findings

Explicit traveling wave solutions are constructed.

Solutions exhibit complex dynamics such as bending.

The Riccati system approach is verified with Mathematica.

Abstract

This work is concerned with the study of explicit solutions for generalized coupled reaction-diffusion and Burgers-type systems with variable coefficients. Including nonlinear models with variable coefficients such as diffusive Lotka-Volterra model, the Gray-Scott model, the Burgers equations. The equations' integrability (via the explicit formulation of the solutions) is accomplished by using similarity transformations and requiring that the coefficients fulfill a Riccati system. We present traveling wave type solutions, as well as solutions with more complex dynamics and relevant features such as bending. A Mathematica file has been prepared as supplementary material, verifying the Riccati systems used in the construction of the solutions.