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

摘要

We present several families of nonlinear reaction diffusion equations withvariable coefficients including Fisher-KPP and Burgers type equations. Specialexact solutions such as traveling wave, rational, triangular wave and N-wavetype solutions are shown. By means of similarity transformations the variablecoefficients are conditioned to satisfy Riccati or Ermakov systems ofequations. When the Riccati system is used, conditions are established so thatfinite-time singularities might occur. The solutions presented containmulti-parameters providing a control on the dynamics of the solutions. In thesuplementary material, we provide a computer algebra verification of thesolutions and exemplify nontrivial dynamics of the solutions.