Where solitons are in a KdV soliton gas

TL;DR

This paper defines soliton positions in a KdV soliton gas using a fluid-cell projection, connecting microscopic soliton dynamics with macroscopic hydrodynamic equations without radiative corrections.

Contribution

It introduces a novel method for identifying soliton positions in dense gases and derives the kinetic equation from first principles, linking integrable systems and hydrodynamics.

Findings

Defined soliton positions via fluid-cell projection.

Derived the kinetic equation for KdV soliton gas.

Established connection with Generalised Hydrodynamics and Bethe equations.

Abstract

The Korteweg-De Vries (KdV) equation is a paradigmatic model of integrable classical fields, admitting solitoning solutions. When many solitons are near to each other, their shapes are modified, and it is not manifest, from the KdV field, where they are. This is a key problem in the analysis of a soliton gas, as its main object, the density of states, is a number of solitons per unit length. How to define solitons' positions at finite densities in the macroscopic limit? A sensible criterium is that, projecting out solitons lying outside a mesoscopic region, the KdV field is unchanged in this region, and the result is a multi-soliton field supported there. In the context of emergent hydrodynamics, this is referred to as a fluid-cell projection. In this paper we solve this problem. We define solitons' positions and a fluid-cell projection, and show that it has these properties, without introducing radiative corrections. We show that the weak limit of conserved densities can be evaluated using the density of states. On large scales the solitons' positions satisfy the semi-classical Bethe equations introduced in the context Generalised Hydrodynamics, that accounts for the two-body scattering shift and encodes factorised scattering. A non-rigorous derivation reproduces the kinetic equation of the KdV soliton gas, first proposed by Gennady El in 2003 using Witham modulation theory from finite-gap solutions. The results hold under simple conditions on spectral parameters, and certain physically natural conditions on impact parameters. No randomness is required. Our proof is based on a novel tau function for the multi-soliton KdV field, which also allows us to obtain new bounds on the growth of the multi-soliton support and on the supremum of the field and its derivatives. We believe the methods are generalisable to other solitonic models.