Critical density of a soliton gas

TL;DR

This paper establishes an upper bound for the density of states in a soliton gas modeled by the KdV equation and determines the speed of sound in such a gas with a Gaussian spectral distribution.

Contribution

It introduces a quantitative measure for the density of a soliton gas and derives the speed of sound within this context, linking quantum and classical soliton dynamics.

Findings

Upper bound for the integrated density of states of the Schrödinger operator

Derived the speed of sound in a Gaussian spectral soliton gas

Connected soliton gas properties with quantum-mechanical operators

Abstract

We quantify the notion of a dense soliton gas by establishing an upper bound for the integrated density of states of the quantum-mechanical Schr\"odinger operator associated with the KdV soliton gas dynamics. As a by-product of our derivation we find the speed of sound in the soliton gas with Gaussian spectral distribution function.