Hydrodynamic type integrable equations on a segment and a half-line

TL;DR

This paper explores integrable boundary conditions for hydrodynamic type systems, providing examples for dispersionless Toda systems, and discusses their relation to Hamiltonian structures on segments and semi-lines.

Contribution

It introduces new integrable boundary conditions for hydrodynamic systems and analyzes their compatibility with Hamiltonian formulations.

Findings

Examples of integrable boundary conditions for dispersionless Toda systems.

Relation between boundary conditions and integrable reductions.

Discussion on Hamiltonian consistency of boundary conditions.

Abstract

The concept of integrable boundary conditions is applied to hydrodynamic type systems. Examples of such boundary conditions for dispersionless Toda systems are obtained. The close relation of integrable boundary conditions with integrable reductions of multi-field systems is observed. The problem of consistency of boundary conditions with the Hamiltonian formulation is discussed. Examples of Hamiltonian integrable hydrodynamic type systems on a segment and a semi-line are presented.