| We are developing Riemann-Hilbert (RH) approach to scattering problems in elastic media. The approach is based on a version of RH method introduced in nineties by A. Fokas for studying boundary problems for linear and integrable nonlinear PDEs. The suitable Lax pair formulation of the elastodynamic equation is obtained. The integral representations obtained from this vector Lax pair are applied to Rayleigh wave propagation in an elastic quarter space and half space. This reduces the problem to the analysis of certain underdetermined matrix RH problem on a torus. We showed that the problem can be in fact re-formulated as a well-posed RH problem with a shift. Some results of the described analysis will be discussed. Part of this work is done jointly with Alexander Its and Julius Kaplunov. |