Invariant Painlevé analysis and coherent structures of two families of reaction-diffusion equations

Exact closed-form coherent structures (pulses/fronts/domain walls) having the form of complicated traveling waves are constructed for two families of reaction–diffusion equations by the use of invariant Painleve analysis. These analytical solutions, which are derived directly from the underlying PDE’s, are investigated in the light of restrictions imposed by the ODE that any traveling wave reduction of the corresponding PDE must satisfy. In particular, it is shown that the coherent structures (a) asymptotically satisfy the ODE governing traveling wave reductions, and (b) are accessible to the PDE from compact support initial conditions. The solutions are compared with each other, and with previously known solutions of the equations.