# Lie symmetries and similarity solutions for rotating shallow water

## TL;DR

This paper analyzes the symmetry properties of the rotating shallow water equations with a polytropic index, deriving invariant solutions and reductions to simpler ODE systems using Lie symmetry methods.

## Contribution

It identifies the Lie symmetry algebra of the system and derives explicit invariant solutions, including travel-wave and scaling solutions, for different values of the polytropic index.

## Key findings

- The system admits a five-dimensional Lie algebra of symmetries.
- Explicit invariant solutions are derived for all reductions.
- Travel-wave and scaling solutions are obtained.

## Abstract

We study a nonlinear system of partial differential equations which describe rotating shallow water with an arbitrary constant polytropic index $\gamma $ for the fluid. In our analysis we apply the theory of symmetries for differential equations and we determine that the system of our study is invariant under a five dimensional Lie algebra. The admitted Lie symmetries form the $\left\{ 2A_{1}\oplus _{s}2A_{1}\right\} \oplus _{s}A_{1}$ Lie algebra for $\gamma \neq 1$ and $2A_{1}\oplus _{s}3A_{1}$ for $\gamma =1$. The application of the Lie symmetries is performed with the derivation of the corresponding zero-order Lie invariants which applied to reduce the system of partial differential equations into integrable systems of ordinary differential equations. For all the possible reductions the algebraic or closed-form solutions are presented. Travel-wave and scaling solutions are also determined.