An inverse problem for linear system of dispersive equations

This paper addresses the inverse problem of identifying the linear velocity coefficient in a linear system governed by two Benjamin-Bona-Mahony-type equations, which model the displacement of water waves propagating along the surface of a shallow channel, incorporating effects of dispersion and topography. To solve this, we reformulate the inverse problem as a restricted minimization problem (RMP), aimed at optimizing a suitably regularized objective functional. We use numerical techniques, specifically the iterative L-BFGS-B algorithm implemented in the Dolfin-Adjoint-Python-SciPy libraries, to solve the RMP effectively. Following methodologies similar to those in Pipicano et al., we establish a local stability result for the RMP. Additionally, through numerical simulations, we demonstrate the effectiveness of the proposed identification method in determining the linear velocity coefficient in Boussinesq-type systems.