Symmetries and exact solutions of the rotating shallow water equations

TL;DR

This paper applies Lie symmetry analysis to the rotating shallow water equations, revealing their symmetry structure, relating them to classical models, and generating new exact, physically interpretable solutions including time-periodic flows.

Contribution

The study identifies the Lie symmetries of the rotating shallow water equations and constructs new exact solutions, including a novel class of time-periodic solutions with quasi-closed particle trajectories.

Findings

Identified a 9-dimensional Lie algebra of symmetries.

Related the rotating shallow water equations to classical models via variable transformations.

Constructed new exact solutions, including time-periodic flows with quasi-closed trajectories.

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.