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

Abstract

We present several families of nonlinear reaction–diffusion equations with variable coefficients including generalizations of 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. We explore solution dynamics across multi-parameters. In the supplementary material, we provide a computer algebra verification of the solutions and exemplify nontrivial dynamics of the solutions.