Constructive asymptotics in the problems of wave splashing on a shallow shore in the framework of nonlinear shallow water equations

The question of the splash of long waves of relatively small amplitude on a gentle shore is being studied. The problem is solved within the framework of a nonlinear system of shallow water equations in a one- or two-dimensional domain. It is assumed that the function D(x,y), which sets the depth of the basin, is smooth, and its gradient does not vanish on the set D=0 (i.e., on the shoreline of the basin in the absence of waves). The smallness of the amplitude is characterized by a small parameter e. One of the main difficulties of the problem is the presence of a free boundary in it, even in the case of non-collapsing waves (this situation is often found in problems about tsunami waves).
To construct asymptotic solutions of the Cauchy problem with small smooth initial data for a nonlinear system of shallow water equations, we use the substitution of variables (such as the simplified Carrier-Greenspan transformation), which depends on the unknown solution itself and transforms the domain in which the latter is defined into an undisturbed domain independent of the solution. Then the resulting nonlinear system is solved by standard methods of perturbation theory. As a zero approximation, a linear hyperbolic system with degeneracy at the boundary of the domain arises. One of the main results of the report is formulated as follows.
Under the above assumptions about the depth function, a nonlinear system of shallow water equations with small initial data has an asymptotic solution up to an arbitrarily high degree of the small parameter e. This asymptotic solution is asymptotically unique. The main term of the asymptotics is constructively expressed through the solution of a linear problem in parametric form.