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Rapidity distribution within the defocusing non-linear Schr\"odinger equation model
Quantum Gases · Submitted 26 Jan 2023
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | background 0.99 | 0.20 | 0.04 | asserted 1.00 |
| s1 | 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. | contribution 0.98 | 0.91 | 0.58 | asserted 0.95 |
| s2 | To do so we use the definition of the rapidity distribution as the asymptotic momentum distribution after an expansion. | method 1.00 | 0.54 | 0.06 | asserted 1.00 |
| s3 | 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. | method 0.97 | 0.92 | 0.07 | asserted 1.00 |
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Original 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.
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2 arXiv papers cite this in 2 sentences
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Cites as background 1 paper
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A very famous asymptotic description of the Lieb-Liniger model is the classical field description which ignores quantization of the particles and describes the system as a classical field $ψ(x)$, where $ψ$ is a complex field and $x$ is the spatial coordinate [ref] .
A comparison between microscopic methods for finite temperature Bose gases · Introduction · 0.94
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