Painlevé test, integrability, and exact solutions for density-dependent reaction–diffusion equations with polynomial reaction functions

A Painlevé test is performed for a general density-dependent reaction–diffusion equation, where the reaction function takes the form of an Nth order polynomial, in order to determine the member models of this class which are integrable. First, we determine the equilibrium behavior for the model. Then, truncated Laurent expansions, relevant to equations having movable branch points at leading order, are used to construct special solutions for the three integrable classes of reaction–diffusion equations which were found. An auto-Bäcklund transformation between two solutions is constructed for an equation having a pole at leading order, which can be used to find further solutions. Some of the solutions are new, and through certain simplifications we may recover old solutions as well.