Nonlinear Effects and Run-up of Coastal Waves Generated by Billiards with Semi-rigid Walls in the Framework of Shallow Water Theory

By coastal waves we mean time-periodic or nearly time-periodic gravity waves on water in a basin of depth $D(x)$, $x=(x_1,x_2)$, that are localized in the vicinity of the coastline $\Gamma ^0=\{D(x)=0\}$. In this paper, for the system of nonlinear shallow water equations, we construct asymptotic solutions corresponding to coastal waves in two specific examples. The solutions are presented in the form of parametrically defined functions corresponding to asymptotic solutions of the linearized system, which, in turn, are related to the asymptotic eigenfunctions of the operator $-\nabla \cdot (g D(x)\nabla )$ that are generated by billiards with semi-rigid walls.