Non-traveling-wave exact solutions for the general reaction-diffusion equation with constant delay

Construct exact solutions beyond traveling wave solutions for the general nonlinear reaction-diffusion partial differential equation with constant delay ut = k u_xx + F(u, ū), where ū = u(x, t − τ), k > 0 is the diffusion coefficient, τ > 0 is a constant delay, and F is an arbitrary real function of two variables.

Background

The paper primarily develops exact solutions for nonlinear Schrödinger equations with delay, but it contrasts these results with the status of reaction-diffusion equations with delay. Equation (3) in the paper defines the general reaction-diffusion PDE with constant delay: ut = k u_xx + F(u, ū), with ū = u(x, t − τ). While traveling wave solutions are known in various contexts, the authors explicitly state that exact solutions of the general form (beyond traveling waves) remain unknown, highlighting a gap in the analytical theory of delayed reaction-diffusion systems.