Nonlocal Hamiltonian structures of the kinetic equation for soliton gas under polychromatic reductions

TL;DR

This paper develops a nonlocal Hamiltonian framework for the kinetic equation of soliton gas under polychromatic reductions, extending previous local models and analyzing specific integrable cases.

Contribution

It introduces a generalized nonlocal Hamiltonian formalism for the kinetic equation, incorporating semi-Riemannian and conformally flat metrics, broadening the scope of prior local models.

Findings

Established nonlocal Hamiltonian structures for soliton gas kinetic equations.

Connected nonlocality to geometric structures like semi-Riemannian metrics.

Analyzed specific integrable cases such as KdV, Lieb-Liniger, and separable models.

Abstract

We deepen the existence of a nonlocal Hamiltonian formalism for the El's kinetic equation for soliton gas under the polychromatic reduction for a class of interaction kernels. The nonlocality presented is related to semi-Riemannian metrics of constant curvature, conformally flat metrics and hypersurfaces in a pseudo-Euclidean space. These results generalise a previous one that Vergallo and Ferapontov obtained with local Hamiltonian operators. Some examples as the Korteweg-de Vries, the Lieb-Liniger and the separable cases are analysed.