Rapidity distribution within the defocusing non-linear Schr\"odinger equation model

TL;DR

This paper derives a relation between conserved quantities and rapidity distribution in the defocusing nonlinear Schrödinger equation, linking classical integrable systems with quantum models through thought experiments.

Contribution

It provides a novel derivation connecting the trace of the monodromy matrix to the rapidity distribution in the classical defocusing nonlinear Schrödinger equation.

Findings

Established a relation between monodromy trace and rapidity distribution.

Presented two different derivations using thought experiments.

Bridged classical integrable systems with quantum Lieb-Liniger model concepts.

Abstract

We consider the classical field integrable system whose evolution equation is the nonlinear Schr\"odinger equation with defocusing non-linearities, which is the classical limit of the quantum Lieb-Liniger model. We propose a simple derivation of the relation between two sets of conserved quantities: on the one hand the trace of the monodromy matrix, parameterized by the spectral parameter and introduced in the inverse-scattering framework, and on the other hand the rapidity distribution, a concept imported from the Lieb-Liniger model. To do so we use the definition of the rapidity distribution as the asymptotic momentum distribution after an expansion. More precisely we use thought experiments implementing an expansion and we present two different ways to derive our result, based on different thought experiments which lead to different calculations.