Riccati-Ermakov systems and explicit solutions for variable coefficient reaction-diffusion equations

TL;DR

This paper derives explicit solutions for variable coefficient reaction-diffusion equations using Riccati-Ermakov systems, including traveling waves and singular solutions, with conditions for finite-time blow-up.

Contribution

It introduces a method to obtain exact solutions for variable coefficient reaction-diffusion equations via Riccati and Ermakov systems, expanding the set of solvable models.

Findings

Explicit solutions include traveling waves, rational, and N-wave types.

Conditions for finite-time singularities are established.

Solutions contain multiple parameters controlling dynamics.

Abstract

We present several families of nonlinear reaction diffusion equations with variable coefficients including Fisher-KPP and Burgers type equations. Special exact solutions such as traveling wave, rational, triangular wave and N-wave type solutions are shown. By means of similarity transformations the variable coefficients are conditioned to satisfy Riccati or Ermakov systems of equations. When the Riccati system is used, conditions are established so that finite-time singularities might occur. The solutions presented contain multi-parameters providing a control on the dynamics of the solutions. In the suplementary material, we provide a computer algebra verification of the solutions and exemplify nontrivial dynamics of the solutions.