Symmetry in Nonlinear Mathematical Physics - 2009

Alexander Chesnokov (Lavrentyev Institute of Hydrodynamics SB RAS, Novosibirsk, Russia)

Lie symmetry analysis of the rotating shallow water equations

Lie group analysis is applied to study nonlinear system that models the finite motion of a rotating shallow liquid contained in a circular paraboloid basin. It is known that the system admits the 9-dimensional Lie algebra of point symmetries (Levi et al., 1987). These symmetries are used to generate new exact solutions of the rotating shallow water equations. In particular, a new class of time-periodic solutions with quasi-closed particle trajectories is constructed and studied. It is shown that the model under consideration is related with the classical shallow water model through the change of variables. This nontrivial transformation allows one to construct and study solutions of the rotating shallow water equations using solutions of the shallow water model and vice versa.