Exact generalized separable solutions to nonlinear delay reaction-diffusion equations

Abstract

Exact generalized separable solutions are derived for the nonlinear delay reaction-diffusion equations $$u_t = ku_{xx} + F(u,w),$$ , where u = u(x, t), w = u(x, t − τ), and τ is the delay time. All of the considered equations depend on one or two arbitrary functions of one argument. The following solutions are found: periodic solutions with respect to time and space variable, solutions that describe the nonlinear interaction between a standing wave and a traveling wave, solutions to generalized Fisher equations with delay, and certain other solutions. Conditions for the instability of some solutions are specified. Exact solutions are also presented for more complex reaction-diffusion equations with several different delay times and equations in which delay arbitrarily depends on the time τ = τ(t). The derived exact solutions contain free parameters (in some cases, there can be any number of these parameters) and can be used to solve certain problems and test approximate analytical and numerical methods for solving these or more complex nonlinear delay equations.