ИСТИНА

Coastal waves are understood as gravitational waves on water in a basin with depth D(x, y), localized in the vicinity of the coastline. This report presents asymptotic solutions of a nonlinear shallow water equations system over a uniformly sloping bottom D(x, y) = γx, describing waves traveling along the shoreline. 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. In the linear problem, the variables are separable, and normal to the coastline the solution is expanded in terms of eigenfunctions of the operator ˆ = −∂2/∂x2− x∂/∂x+ (k2x−ω2/γ) with boundedness conditions at the shoreline |ξ|x=0 <∞ and decay at infinity. The connection of the constructed solution with the classical (integrable) “billiard with a semi-rigid wall” is also discussed. This work was supported by the Russian Science Foundation under grant no. 24-11-00213.