Inverse obstacle scattering for elastic waves in the time domain

This paper concerns an inverse elastic scattering problem which is to determine a rigid obstacle from time domain scattered field data for a single incident plane wave. By using the Helmholtz decomposition, we reduce the initial-boundary value problem for the time domain Navier equation to a coupled initial-boundary value problem for wave equations, and prove the uniqueness of the solution for the coupled problem by employing the energy method. The retarded single layer potential is introduced to establish a set of coupled boundary integral equations, and uniqueness is discussed for the solution of these boundary integral equations. Based on the convolution quadrature method for time discretization, the coupled boundary integral equations are reformulated into a system of boundary integral equations in the s-domain, and then a convolution quadrature based nonlinear integral equation method is proposed for the inverse problem. Numerical experiments are presented to show the feasibility and effectiveness of the proposed method.