On the solution of an inverse problem for equations of shallow water in a pool with variable depth

We consider the problem of propagation of small-amplitude waves on the surface of shallow water in a reservoir with variable depth. From the system of shallow water equations, we derive the Korteweg-de Vries equation (KdV) with a variable coefficient that takes into account both the bottom profile and the geometric divergence of the waves. The inverse problem, which consists of determining the variable profile of the bottom by the period and amplitude of stationary waves is posed and solved within the adiabatic approximation. It is shown that taking into account the geometric divergence of waves significantly affects the result of solving the inverse problem.