Eigenfunctions of structures described by the “shallow water” model in a plane

We propose a new method for solving the “shallow water” equations. We show that from the equations of the “shallow water” model one obtains nonlinear Liouville type equations, Helmholtz equations, etc. This allows one to construct eigenfunctions of various structures that appear in the flow in the two-dimensional case. We obtain exact and asymptotic solutions in an elliptic domain with singularities.