Asymptotics of Long Nonlinear Coastal Waves in Basins with Gentle Shores

Abstract

We construct asymptotic solutions of a special type for the nonlinear system of shallow water equations in two-dimensional basins with gentle shores and depth function \(D(x)\) , where \(x=(x_1,x_2)\) . These solutions represent waves localized near the shorelines (coastal waves) and generalize the (linear) Stokes and Ursell waves. The waves we consider are periodic or close to periodic in time. The corresponding asymptotic solutions are represented in a parametric form based on the modification of the Carrier–Greenspan transformation and are generated by asymptotic eigenfunctions (quasimodes) of the operator \(\hat{H} = -\nabla\cdot(gD(x)\nabla)\) , where \(g\) is the gravity acceleration. These eigenfunctions are, in general, related to the trajectories of a Hamiltonian system with the Hamiltonian \(H = gD(x)(p_1^2+p_2^2)\) , which forms billiards with “semi-rigid walls.” In the general case, the existence of such billiards assumes the integrability condition that is practically impossible to be satisfied in real situations. However, we consider a “degenerate” situation where the trajectories are localized in a very narrow vicinity of the boundary \(\Gamma_0=\{D(x)=0\}\) , and the asymptotic eigenfunctions resemble the well-known “whispering gallery” wave functions in acoustics. In this case, the requirement of integrability is eliminated (the corresponding billiard is “almost integrable” for the considered set of trajectories). One important difference between the problem we study and the classical whispering gallery situation is that, due to the degeneracy of the depth function \(D(x)\) on the boundary \(\Gamma_0\) , the trajectories are always normal to the boundary, and the requirement of convexity of the domain of the considered problem is absent.

DOI 10.1134/S106192084010060