ИСТИНА

By coastal waves we understand gravitational waves on water in a basin with depth D(x,y), localized in the vicinity of the coastline G = {D(x,y) = 0}. In this report the Cauchy problem for 2D shallow water system in the case of sloping bottom D(x,y) = x is considered and asymptotic formulas for traveling coastal waves are presented. The asymptotic solutions of the nonlinear shallow water equations system are written in the form of parametrically defined functions determined through exact solutions of the linearized system (see [Dobrokhotov, Minenkov, Nazaikinskii, 2022] for general linearization procedure and [Dobrokhotov, Minenkov, Votiakova, 2024] for coastal waves asymptotics that are harmonic with time). In the linear problem, even for a general bottom function, the variables can be separated in adiabatic approximation in the form of operator separation of variables. The reduced 1D equation along the shoreline has the dispersion relation ω2 ∼ k, which results in dispersion effects. The relation of the constructed asymptotics with the classical (integrable) "billiard with a semi-rigid wall" (see [Dobrokhotov, Nazaikinskii, Tsvetkova, 2023], [Bolotin, Treshev, 2024]) is also discussed. The results are obtained together with S.Yu.Dobrokhotov and M.M.Votiakova and the report is done with the financial support of the Ministry of Science and Higher Education of the Russian Federation in the framework of a scientific project under agreement 075-15-2025-013.