Periodic traveling waves in a nonlocal erosion equation

Abstract

We consider a periodic boundary-value problem for a nonlinear partial differential equation containing terms with a deviating spatial argument. The functional-differential equation under consideration was previously proposed as a model for describing the process of relief formation on a surface of semiconductors under ionic bombardment. We show that the boundary-value problem under consideration can have an asymptotically large number of two-dimensional invariant manifolds formed by solutions that have the structure of traveling periodic waves. We also show that these invariant manifolds are typically saddle ones, and the number of those that are local attractors does not exceed two. We obtain asymptotic formulas for solutions belonging to a given invariant manifolds. These mathematical results partially explain the complexity of dynamics of pattern formation on the surface of semiconductors.