Periodic Contrast Structures in the Reaction–Diffusion–Advection Equation with a KPZ Nonlinearity

A singularly perturbed time-periodic boundary-value problem for a parabolic reaction–advection–diffusion equation with a nonlinearity containing the squared gradient of the unknown function (KPZ nonlinearity) is studied. A periodic solution with an internal transition layer is considered in the noncritical and critical cases. An asymptotic approximation of the solution is constructed, and the asymptotic behavior of a point of the transition layer is determined. Existence theorems and asymptotic stability are proved by the method of differential inequalities.