ИСТИНА

By coastal waves we mean periodic or near-periodic gravitational waves on water in a basin of depth D(x), x = (x1, x2), localized in the vicinity of the coastline Γ0 = {D(x) = 0}. specific examples, we construct asymptotic solutions of a system of nonlinear shallow water equations corresponding to shore waves in the form of parametrically defined functions defined through the asymptotics of a linearized system, which in turn are related to the asymptotic eigenfunctions of the operator L = −∇gD(x)∇. The definitions of the operator are smooth functions ξ(x) in the domain Ω = {x : D(x) > 0} with finite energy: |ξ|x ∈ Γ0 < ∞. The connection of the constructed asymptotics with classical (almost integrable) "billiards with semi-rigid walls" is also discussed. The work was carried out with the financial support of the RGNF grant 24-11-00213.