Where solitons are in a KdV soliton gas

Abstract

The Korteweg-De Vries (KdV) equation, a paradigmatic integrable model, admits multi-soliton solutions where localised profiles interact by elastic, factorised scattering. But when solitons interact, their shapes are modified, and individual solitons cannot be identified anymore, a key problem in soliton gases where one needs a density of solitons per unit distance. This is the question of quasi-particles in integrable systems: can we locate asymptotic objects at finite times and densities in the macroscopic limit of infinite number of solitons? A good definition of solitons' positions should guarantee that the field in any mesoscopic neighbourhood is determined by solitons lying there. We solve this problem for the KdV equation. We define solitons' positions as explicit solutions to the semiclassical Bethe equations. We show a density lower bound generalising the hard-rod condition and reminiscent of the condensate lower-bound known from soliton gases. We construct a fluid-cell projection, that projects out solitons lying outside mesoscopic regions, and show that the result is a field composed solely of the remaining solitons, unchanged in this region and exponentially supported there. From this we show that in the weak limit, conserved densities are given by the empirical density for these solitons' positions. Thus we have associated a kinetic theory of quasi-particles to the KdV equation. We also find new bounds on the growth of the multi-soliton support and on the supremum of the field and its derivatives. The results hold under natural conditions on spectral and impact parameters, and no randomness or large-wavelength conditions are required. Our proof is based on a novel solitonic tau function. This is a first step towards rigorously deriving the kinetic equation of the KdV soliton gas, first proposed by Gennady El in 2003. The methods are generalisable to other solitonic models.