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On eigenfunctions of the structures described by the “shallow-water” model on the plane. (English) Zbl 1151.76396
J. Math. Sci., New York 151, No. 1, 2639-2650 (2008); translation from Fundam. Prikl. Mat. 12, No. 6, 17-32 (2006).
Summary: We propose a new method for solving the “shallow-water” equations. We show that from the equations of “shallow water” 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.
MSC:
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q35 PDEs in connection with fluid mechanics
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