Kinetic equation for soliton gas: integrable reductions

TL;DR

This paper develops an integrable kinetic equation model for soliton gases, extending hydrodynamic theory to systems with Jordan block structures and demonstrating their solvability via a generalized hodograph method.

Contribution

It introduces a novel integrable kinetic equation for soliton gases with non-diagonalisable hydrodynamic systems, expanding Tsarev's theory to include Jordan block structures.

Findings

Demonstrated integrability of the non-diagonalisable hydrodynamic system.

Established a hierarchy of commuting hydrodynamic flows.

Extended the generalized hodograph method to systems with Jordan blocks.

Abstract

Macroscopic dynamics of soliton gases can be analytically described by the thermodynamic limit of the Whitham equations, yielding an integro-differential kinetic equation for the density of states. Under a delta-functional ansatz, the kinetic equation for soliton gas reduces to a non-diagonalisable system of hydrodynamic type whose matrix consists of several 2x2 Jordan blocks. Here we demonstrate the integrability of this system by showing that it possesses a hierarchy of commuting hydrodynamic flows and can be solved by an extension of the generalised hodograph method. Our approach is a generalisation of Tsarev's theory of diagonalisable systems of hydrodynamic type to quasilinear systems with non-trivial Jordan block structure.