Exact and asymptotic solutions of a system of nonlinear shallow water equations in basins with gentle shores

We suggest an effective approximate method for constructing solutions to problems with a free boundary for 1-D and 2-D-systems of nonlinear shallow water equations in basins with gentle shores. The method is a modification (and pragmatic simplification) of the Carrier-Greenspan transformation in the theory of 1-D shallow water over a flat sloping bottom. The result is as follows: approximate solutions of nonlinear equations are expressed through solutions of naively linearized equations via parametrically defined functions. This allows us to describe the effects of waves run-up on a shore and their splash. Among the applications we can mention tsunami waves, seiches and coastal waves. We also present a comparison of the obtained formulas with the V.A. Kalinichenko (Institute for Problems in Mechanics RAS) experiment with standing Faraday waves in an extended basin with gently sloping shores.
Joint work with D. Minenkov.