A new method for constructing exact solutions to nonlinear delay partial differential equations

Abstract

We propose a new method for constructing exact solutions to nonlinear delay reaction--diffusion equations of the form $$ u_t=ku_{xx}+F(u,w), $$ where $u=u(x,t)$, $w=u(x,t-\tau)$, and $\tau$ is the delay time. The method is based on searching for solutions in the form $u=\sum^N_{n=1}\xi_n(x)\eta_n(t)$, where the functions $\xi_n(x)$ and $\eta_n(t)$ are determined from additional functional constraints (which are difference or functional equations) and the original delay partial differential equation. All of the equations considered contain one or two arbitrary functions of a single argument. We describe a considerable number of new exact generalized separable solutions and a few more complex solutions representing a nonlinear superposition of generalized separable and traveling wave solutions. All solutions involve free parameters (in some cases, infinitely many parameters) and so can be suitable for solving certain problems and testing approximate analytical and numerical methods for nonlinear delay PDEs. The results are extended to a wide class of nonlinear partial differential-difference equations involving arbitrary linear differential operators of any order with respect to the independent variables $x$ and $t$ (in particular, this class includes the nonlinear delay Klein--Gordon equation) as well as to some partial functional differential equations with time-varying delay.