Kinetic equation for a soliton gas and its hydrodynamic reductions

TL;DR

This paper introduces a new class of kinetic equations modeling soliton gases, proves their integrability through hydrodynamic reductions, and provides explicit solutions and symmetries, advancing understanding of nonlinear wave dynamics.

Contribution

The paper develops a novel kinetic equation for soliton gases, demonstrates its integrability via hydrodynamic reductions, and derives explicit Riemann invariants and solutions.

Findings

Hydrodynamic reductions are integrable linearly degenerate systems.

Explicit Riemann invariants and characteristic velocities are derived.

Exact solutions exhibit quasi-periodic and self-similar properties.

Abstract

We introduce and study a new class of kinetic equations, which arise in the description of nonequilibrium macroscopic dynamics of soliton gases with elastic collisions between solitons. These equations represent nonlinear integro-differential systems and have a novel structure, which we investigate by studying in detail the class of $N$-component `cold-gas' hydrodynamic reductions. We prove that these reductions represent integrable linearly degenerate hydrodynamic type systems for arbitrary $N$ which is a strong evidence in favour of integrability of the full kinetic equation. We derive compact explicit representations for the Riemann invariants and characteristic velocities of the hydrodynamic reductions in terms of the `cold-gas' component densities and construct a number of exact solutions having special properties (quasi-periodic, self-similar). Hydrodynamic symmetries are then derived and investigated. The obtained results shed the light on the structure of a continuum limit for a large class of integrable systems of hydrodynamic type and are also relevant to the description of turbulent motion in conservative compressible flows.