Multidimensional and surface solitons in a nonlinear elastic medium

Nonlinear shear waves in two-dimensional systems (in particular, surface waves) are investigated with allowance for the spatial dispersion of the elastic medium. It is shown that the dispersion plays an important role in the structural and modulational stability of the nonlinear waves and to a large degree determines the directions of localization of phonons in a nonlinear localized wave and, in particular, the possibility of existence of elastic surface solitons. By means of an asymptotic procedure, solutions are found for small-amplitude two-dimensional elastic shear solitons of the one-parameter stationary-profile type and for envelope solitons and also for surface solitons localized near an ideal surface of an elastic half space. Localized excitations of this kind can exist only in a medium with a “focusing” (soft) nonlinearity and positive dispersion ∂ 2 ω/∂k 2 >0, where ω(k) is the dispersion relation for linear waves. A procedure is proposed for finding solutions for surface envelope solitons localized near a surface covered with a layer of another substance. A comparison is made between the structures of the surface shear solitons at an ideal surface and at a surface with a film coating.