Classical and wave dynamics of long nonlinear coastal waves, localized near gentle shores

In recent works [S.Yu.Dobrokhotov, V.E.Nazaikinskii, A.V.Tsvetkova, Proc. Steklov Inst. Math., 2023] and [S.Yu.Dobrokhotov, D.S.Minenkov, M.M.Votiakova, Russ. J. Math. Phys., 2024], two-dimensional coastal waves were studied and corresponding asymptotic solutions for nonlinear shallow water system were constructed. (By coastal waves we understand nonlinear waves that are localized near the shorelines and generalize the (linear) Stokes and Ursell waves.) In the current paper, we present asymptotic formulas for coastal waves in more practically suitable coordinates, discuss relations between parameters and typical properties of coastal waves (incl. non-breaking conditions for amplitude) and consider nontrivial examples in non-convex basins. We also discuss relations of constructed solutions to trajectories of a Hamiltonian system, which coefficients degenerate on the boundary of the considered domain and which system can be studied in the framework of fast and slow variables. Such trajectories correspond to "degenerate billiards with semi-rigid walls", that were studied in the general case in the recent work [S.Bolotin, D.Treschev, Another Billiard Problem, Russ. J. Math. Phys., 2024].