Theory of Nonlinear Surface Waves and Solitons

Abstract

In this contribution we explore the possibilities of existence of various types of nonlinear surface waves that may propagate in a half-space of elastic materials. Nonlinearities may be of physical and geometrical origins. Various types of surface waves and exemplary nonlinear wave equations are first presented. Then basic equations of nonlinear elasticity theory are introduced and the standard linear surface-wave problems are recalled, in particular the one governing Rayleigh waves. Then two large classes of problems are examined in the presence of nonlinearities, whether there exists no dispersion or in the presence of dispersion. In the first case, various methods leaning on asymptotics are presented; these are the so-called iterative, projection, and Hamiltonian methods. This allows one to describe the evolution of nonlinear waves as they propagate, as also some exceptional cases of waves of permanent form. In the second case compensation between nonlinearity and dispersion may favor the existence of surface solitary waves of various types including surface SH envelope solitons guided by a thin film glued on the nonlinear substrate, as also solitary Rayleigh waves, with an emphasis on nonlocal effects. Generalizations including propagation along plates are given in the conclusion.