Darboux integrability for diagonal systems of hydrodynamic type

TL;DR

This paper characterizes Darboux integrability for diagonal hydrodynamic systems, linking it to commuting flows, Laplace transformations, and geometric structures, and provides new insights and examples in this context.

Contribution

It establishes equivalences between Darboux integrability, commuting flows, and Laplace transformation termination, and offers geometric interpretations and new examples.

Findings

Darboux integrability is equivalent to the integrability of commuting flows.

Termination of Laplace transformation sequences characterizes Darboux integrability.

Darboux integrable systems are necessarily semihamiltonian.

Abstract

We prove that 1) diagonal systems of hydrodynamic type are Darboux integrable if and only if the corresponding systems for commuting flows are Darboux integrable, 2) systems for commuting flows are Darboux integrable if and only if the Laplace transformation sequences terminate, 3) Darboux integrable systems are necessarily semihamiltonian. We give geometric interpretation for Darboux integrability of such systems in terms of congruences of lines and in terms of solution orbits with respect to symmetry subalgebras, discuss known and new examples.