The aim of this work is to prove the existence and uniqueness theorems of the solution of an inverse problem and associated with them an initial boundary-value problem for a nonlinear model describing the process of coupling of electromagnetic and elastic waves. We start by introducing a simple model consisting from two differential equations, one of them is a hyperbolic equation (an analog of the Lamé system) and another one is a parabolic equation (an analog of the diffusion approximation of Maxwell's system) coupled by nonlinear terms in both equations. An application of this result can be found in explanation of nonlinear processes in the theory of electro-magnetoelastic interactions in continuum physics and geophysics.