Exact equilibrium solutions of a diffusion equation with a nonlinear diffusion term by means of Jacobian elliptic functions

Abstract

We construct exact equilibrium solutions of a diffusion equation with a nonlinear diffusion term by means of Jacobian elliptic functions, the direct method, and an f 4 model through a simple mapping relation. In the limit, as the elliptic modulus of the Jacobian elliptic function tends to 0 and to 1, we deduce the trigonometric and solitary wave solutions, respectively. The long-time behaviour of solutions is investigated via numerical simulations.