Hydrodynamic-type systems describing 2-dimensional polynomially integrable geodesic flows

TL;DR

This paper derives multi-component hydrodynamic-type systems from polynomial integrals that describe 2D geodesic flows, demonstrating their integrability and providing a constructive method to generate conservation laws.

Contribution

It introduces a systematic way to obtain integrable hydrodynamic systems from polynomial integrals of geodesic flows, including particular cases based on root multiplicity.

Findings

All systems are semi-Hamiltonian and integrable via the generalized hodograph method.

Polynomial first integrals enable construction of conservation law generating equations.

Classification of integrable cases according to polynomial root multiplicity.

Abstract

Starting from a homogeneous polynomial in momenta of arbitrary order we extract multi-component hydrodynamic-type systems which describe 2-dimensional geodesic flows admitting the initial polynomial as integral. All these hydrodynamic-type systems are semi-Hamiltonian, thus implying that they are integrable according to the generalized hodograph method. Moreover, they are integrable in a constructive sense as polynomial first integrals allow to construct generating equations of conservation laws. According to the multiplicity of the roots of the polynomial integral, we separate integrable particular cases.