Hodograph method for solving the overturned shallow water problem

The behavior of overturned shallow water (a layer of water on the ceiling) described by a system of two first-order quasilinear partial differential equations is studied using the hodograph method based on a conservation law. The basic difference of these equations from the classical shallow water ones is that the force of gravity varies in direction. It is assumed that the fluid layer is “glued” to the horizontal solid surface and the gravity is directed away from the surface. As a result, the equations become elliptic. The considered evolution Cauchy problem is a model describing an unstable continuous medium of the quasi-Chaplygin gas type. By applying the developed method, the evolution Cauchy problem for the system of two first-order quasilinear partial differential equations is transformed into the Cauchy problem for a system of ordinary differential equations. The behavior of a stationary plane fluid layer subject to a spatially periodic smooth perturbation is considered as an example. It is shown that a nonsmooth spatially periodic structure representing a standing cnoidal wave develops on the fluid surface over a finite time interval.