This article is cited in 26 papers

Symmetries and exact solutions of the shallow water equations for a two-dimensional shear flow

A. A. Chesnokov

Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk, 630090, Russia

Abstract: This paper considers nonlinear equations describing the propagation of long waves in two-dimensional shear flow of a heavy ideal incompressible fluid with a free boundary. A nine-dimensional group of transformations admitted by the equations of motion is found by symmetry methods. Two-dimensional subgroups are used to find simpler integrodifferential submodels which define classes of exact solutions, some of which are integrated. New steady-state and unsteady rotationally symmetric solutions with a nontrivial velocity distribution along the depth are obtained.

Keywords: symmetry, exact solutions, two-dimensional shear flows, long waves.

UDC: 532.591+517.958

Received: 22.06.2007

 English version: Journal of Applied Mechanics and Technical Physics, 2008, 49:5, 737–748

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