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Special classes of solutions to the equations of horizontal-displacement fluid motion. (Russian) Zbl 1249.76040
Summary: We construct a special class of solutions to the nonlinear integrodifferential equations describing the horizontal displacement motion of an ideal fluid in an open channel in the shallow water approximation. This class of solutions is characterized by a linear relation among Riemann integral invariants and is determined by a hyperbolic system of two differential equations with two parameters. In the class of travelling waves, we construct exact solutions, including arbitrary functions, to the equations of motion continuously adjacent to a given displacement flow.

76E05 Parallel shear flows in hydrodynamic stability
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction