Classical and Wave Dynamics of Long Nonlinear Waves Localized in the Vicinity of Gently Sloping Shores

Asymptotic solutions corresponding to coastal waves for a two-dimensional nonlinear system of shallow water equations were constructed in recent papers (2023, 2024) by S. Yu. Dobrokhotov and coauthors. In the present paper we derive asymptotic formulas for nonlinear coastal waves in coordinates more convenient for specific situations, study the properties of nonlinear waves, including the relations between the amplitude and wavelength of nonbreaking solutions, and consider meaningful examples. In addition, we discuss the relationship between the constructed solutions and the trajectories of a Hamiltonian system with coefficients degenerating on the boundary of the domain in which fast and slow variables can be introduced. Such trajectories form “degenerate billiards with semi-rigid walls,” which were studied in a more general case by S. V. Bolotin and D. V. Treschev (2024).