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

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. Twodimensional 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.Introduction. Approximate models of shallow-water theory are used to model wave processes in fluids and to describe large-scale motions in the atmosphere and ocean. A mathematical foundation for the classical (depth-averaged) shallow-water approximation was given by Ovsyannikov [1]. The long-wave model taking into account velocity shear along the depth, especially in the two-dimensional case has been studied to a lesser extent. The nonlinear equations of rotational shallow water for plane-parallel motions were studied in [2-7], etc., where infinite series of conservation laws were found, classes of exact solutions were constructed, and conditions for the generalized hyperbolicity and well-posedness of the Cauchy problem were formulated. Teshukov [8,9] studied the shallow-water equations for two-dimensional shear flow, established the existence of simple waves, constructed an extension of Prandtl-Meyer waves, and formulated conditions for the generalized hyperbolicity of the steady-state equations.In the present work, a theoretical group analysis of the two-dimensional shallow-water equations for shear flows was performed. The 9-dimensional group of the admitted transformations was found. It was established that the Lie algebra of operators L 9 corresponding to these transformations is isomorphic to the Lie algebra of the admitted operators for the equations of two-dimensional isentropic motion of a polytropic gas with an adiabatic exponent γ = 2, for which the optimal system subalgebras [10] is known. New classes of exact solutions were constructed using an optimal system of subalgebras that allows a classification of submodels. Steady-state rotationally symmetric solutions describing motion with zones of return flow were obtained. Stable unsteady shear solutions describing the spread (collapse) of a parabolic cavity were found.1. Mathematical Model and Admitted Transformations. The solutions of the system of differential equations

Hamiltonian formalism of two-dimensional Vlasov kinetic equation

“…Such very important problems as well-posedness and nonlinear stability of the Benney-type model obtained by the Heaviside closure were investigated in [21,22]. Another paper [23] was dedicated to the investigation of the Benney system (1.3) from the Lie group analysis. Interesting classes of particular solutions were found in this paper.…”

“…The author thanks Alexander Chesnokov, Sergey Gavrilyuk, Oleg Mokhov, Vladimir Taranov, Sergey Tsarev and Victor Vedenyapin for their stimulating and clarifying discussions. In addition, the author expresses his gratitude to both anonymous referees, who pointed out the results obtained by other researchers working in the same field (see [15,[21][22][23][24]) and who improved the quality and clarity of this paper. …”

Shallow water equations on a rotating attracting sphere 2. Simple stationary waves and sound characteristics

“…In the present paper, we describe sound characteristics on the equilibrium state and study stationary simple waves of system (1.1). We also note papers [3,4], in which the group properties of some models of atmospheric physics are studied and exact solutions constructed using the symmetry groups admitted by these models are studied.…”

Shallow‐water equations with complete Coriolis force: Group properties and similarity solutions

“…There are various applications of the Lie point symmetries in physical theories 15‐22 as also in fluid dynamics and in shallow‐water equations. The Lie point symmetries of the nonrotating shallow‐water equations were studied in Chesnokov, 23 while a similar analysis with specific Coriolis term was performed in Chesnokov 24 . In the latter case, it was found that the special Coriolis term can be neglected by the field equations with the use of point transformations.…”