Hamiltonian systems of Jordan block type: delta-functional reductions of the kinetic equation for soliton gas

TL;DR

This paper investigates the Hamiltonian structures of Jordan block type systems related to soliton gases, establishing conditions for their multi-Hamiltonian formulations in various integrable models.

Contribution

It identifies linear degeneracy as essential for Hamiltonian structures and develops multi-Hamiltonian formulations for delta-functional reductions of the kinetic equation.

Findings

Linear degeneracy is necessary for Hamiltonian structures.

Multi-Hamiltonian formulations are established for several integrable models.

Applicable to KdV, sinh-Gordon, and other soliton-related systems.

Abstract

We demonstrate that linear degeneracy is a necessary condition for quasilinear systems of Jordan block type to possess first-order Hamiltonian structures. Multi-Hamiltonian formulation of linearly degenerate systems governing delta-functional reductions of the kinetic equation for dense soliton gas is established (for KdV, sinh-Gordon, hard-rod, Lieb-Liniger, DNLS, and separable cases).