Zeroth-order conservation laws of two-dimensional shallow water equations with variable bottom topography

TL;DR

This paper classifies zeroth-order conservation laws for two-dimensional shallow water equations with variable bottom topography, using an optimized method and relating findings to Hamiltonian structures and Lie symmetries.

Contribution

It provides a comprehensive classification of conservation laws for these equations, including new point equivalences and their relation to Lie symmetries and Hamiltonian structures.

Findings

Classification of conservation laws up to equivalence.

Identification of additional point equivalences.

Construction of generating sets under Lie symmetries.

Abstract

We classify zeroth-order conservation laws of systems from the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The classification is carried out up to equivalence generated by the equivalence group of this class. We find additional point equivalences between some of the listed cases of extensions of the space of zeroth-order conservation laws, which are inequivalent up to transformations from the equivalence group. Hamiltonian structures of systems of shallow water equations are used for relating the classification of zeroth-order conservation laws of these systems to the classification of their Lie symmetries. We also construct generating sets of such conservation laws under action of Lie symmetries.