Uniqueness of non-monotone traveling waves for delayed reaction-diffusion equations

This work is concerned with the traveling wave solutions in a class of delayed reaction–diffusion equations with crossing-monostability. In a previous paper, we established the existence of non-monotone traveling waves. However the problem of whether there can be two distinct traveling wave solutions remains open. In this work, by rewriting the equation as an integral equation and using the theory on nontrivial solutions of a convolution equation, we show that the non-monotone traveling waves are unique up to translation. We also obtain the exact asymptotic behavior of the profile as ξ→−∞ξ→−∞ and the conditions of non-existence of traveling wave solutions.