Asymptotic Solutions of the Cauchy Problem for the Nonlinear Shallow Water Equations in a Basin with a Gently Sloping Beach

Abstract

Small amplitude solutions of the nonlinear shallow water equations in a one- or two-dimensional domain are considered. The amplitude is characterized by a small parameter $\varepsilon$. It is assumed that the basin depth is a smooth function whose gradient is nowhere zero on the set of its zeros (i.e., on the coastline in the absence of waves). A solution of the equations is understood to be a triple (time-dependent domain, free surface elevation, horizontal velocity) smoothly depending on $\varepsilon$ and such that (i) the free surface elevation and the horizontal velocity are zero for $\varepsilon =0$; (ii) the sum of the free surface elevation and the depth is positive in the domain and zero on the boundary; (iii) the free surface elevation and the horizontal velocity are smooth in the closed domain and satisfy the equations there. An asymptotic solution modulo $O( \varepsilon ^N)$ is defined in a similar way except that the equations must be satisfied modulo $O( \varepsilon ^N)$. We prove that, in this setting, the nonlinear shallow water equations with small smooth initial data have an asymptotically unique asymptotic solution modulo $O( \varepsilon ^N)$ for arbitrary $N$. The proof is constructive (and leads to simple explicit formulas for the leading asymptotic term). The construction uses a change of variables (depending on the unknown solution and resembling the Carrier–Greenspan transformation) that maps the unknown varying domain onto the unperturbed domain. The resulting nonlinear system is within the scope of regular perturbation theory. The zero approximation is a Cauchy problem for a linear hyperbolic system with degeneracy on the boundary, whose unique solvability in the class of smooth functions is proved by lifting the problem to a closed 3-manifold (where the spatial part of the operator turns out to be hypoelliptic).