Surface solitons in nonlinear elastic media

We investigate nonlinear surface shear waves in the presence of spatial dispersion and demonstrate that this dispersion plays a great role in the structure and modulation stability of nonlinear surface waves. We show that the possibility of an existence of surface solitons is connected with the dispersion properties of crystal. Using an asymptotic procedure we found the small amplitude solution for the nonlinear surface waves as a power series with the expansion parameter (omega(k)-omega)/chi k(2), where omega(k)={k(2) + chi k(4)}(1/2) is the dispersion law of linear waves. In the dispersive media with ''focusing'' nonlinearity the nonlinear surface shear waves are modulationally stable in the case chi<0. But for chi>0 these waves are modulationally unstable and give rise to the elastic surface solitons, localized in the plane of the surface. The amplitude of this envelope soliton is proportional to the value {omega(k)-omega}(1/2)/k(2), its velocity is dose to the group velocity of phonons with the same wave number, the region of localization near the surface is of the order {omega(k)-omega}(-1/2) and the size of the localization region in the surface plane is k chi(1/2){omega(k)-omega}(-1/2). Hence, the surface solitons exist only in the elastic media with the positive dispersion (d(2) omega/dk(2)>0).