Symmetries and exact solutions of the rotating shallow-water equations

Abstract

Abstract. Lie symmetry analysis is applied to study the nonlinear rotating shallow water equations. The 9-dimensional Lie algebra of point symmetries admitted by the model is found. It is shown that the rotating shallow water equations are related with the classical shallow water model with the change of variables. The derived symmetries are used to generate new exact solutions of the rotating shallow equations. In particular, a new class of time-periodic solutions with quasi-closed particle trajectories is constructed and studied. The symmetry reduction method is also used to obtain some invariant solutions of the model. Examples of these solutions are presented with a brief physical interpretation.

Ermakov–Ray–Reid Systems in (2+1)‐Dimensional Rotating Shallow Water Theory

“…It is interesting to remark that the constraint (A.1) corresponds to a privileged class of motions that may be connected to irrotational shallow water motions with f = 0. Thus, Chasnokov [39] recently used Lie-group analysis to establish a novel connection between the rotating shallow water system (1)-(3) with Z * = 0 and an associated nonrotating system with f = 0. The general Lie-group analysis of [6] may be readily adduced to extend this result to elliptic paraboloidal bottom topographies.…”

Hamiltonian formalism of two-dimensional Vlasov kinetic equation

“…Interesting classes of particular solutions were found in this paper. One of these solutions determined by the so-called Dirac closure (4) was also studied by methods of Lie group analysis in [24]. The derived symmetries were used to generate exact solutions of the rotating shallow water equations (4).…”

“…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. …”

One-Dimensional Optimal System for 2D Rotating Ideal Gas

“…whereF = h γ − (a − u) 2 h and z = t − αx. Because we performed reduction with a subalgebra admitted by the nonrotating system, by setting f = 0 in (20), (21), (22) we get the similarity solution for the nonrotating system where in this case is found to be h (z) = h 0 , u (z) = u 0 and v (z) = v 0 .…”