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Rapidity distribution within the defocusing non-linear Schrödinger equation model

by Yasser Bezzaz, Léa Dubois, Isabelle Bouchoule

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Submission summary

Authors (as registered SciPost users): Isabelle Bouchoule
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-07-06
Date submitted: 2023-05-22 14:38
Submitted by: Bouchoule, Isabelle
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
  • Atomic, Molecular and Optical Physics - Theory
Approach: Theoretical


We consider the classical field integrable system whose evolution equation is the non-linear 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 a very large expansion. We propose two different ways to derive the result, each one using a thought experiment that implements an expansion.

Published as SciPost Phys. Core 6, 064 (2023)

Author comments upon resubmission

Dear editor, dear referees,

We thank the referees for their work on our manuscript and their intersting remarks. Please see the answer we provide on the Scipost webpage to each referee report. Each point raised by the referee has been considered and the paper has been modified accordingly.

We hope the referees will agree with our replies,

Best regards,
The authors.

List of changes

- Last sentence of the abstract has been modified to be more clear
-The beginning of the introduction has been modified and expanded. We emphasize the importance of the Lieb-Liniger model both for its experimental relevance and its theoretical key role. We added corresponding references.
-In the introduction, we replaced "homogeneous to a momentum" by "whose unit is mass x velocity "
-Before Eq. 4, we added "As explained in the introduction ...". After the equation, we added a sentence to emphasize we use EQ. 4 as the definition of the rapidity distribution.
-After Eq. (5), we extended the discussion to compare to previously published results. In particular, we discuss the link with the results of the references [A. D. Luca and G. Mussardo, J. Stat. Mech. 064011 (2026)] and [G. del Vecchio del Vechio et al., Scipost Physics 9, 002 (2022)].
-At the beginning of section 4, we recall that we consider a field with periodic boundary conditions on a box of length L.
-Before Eq. (6), we added the sentence "Thus in the following we consider the one-dimensional function $x\rightarrow \psi(x,t)$ and we omit the time variable."
-We modified the second paragraph of the introduction. We now write that "Eq. (5) has previously been derived using more mathematical approaches". We also mention at the end of this paragraph that one could explore the possibility to use the link between the rapidity distribution and the dynamical structure factor to provide an alternative calculation of Eq. (5).
-We added a new paragraph to the conclusion that makes the link between the thought experiments investigated in this paper and quenches protocols previously considered within the Lieb-Liniger model.
-We made several corrections of English, following the advises of the referees.

Reports on this Submission

Anonymous Report 3 on 2023-6-15 (Invited Report)


I'm satisfied with the improvements made by the authors and recommend the manuscript for the publication.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 2 on 2023-6-2 (Invited Report)


Same as previous report


the authors fulfilled all my requests, the paper can now be published

  • validity: high
  • significance: high
  • originality: top
  • clarity: high
  • formatting: excellent
  • grammar: excellent

Anonymous Report 1 on 2023-5-23 (Invited Report)


Same as on my first report


Same as in first report except that the authors have made improvements following my suggestions. Most of my criticism was about the paper not being very clear in terms of writing style. This has now been improved.


The authors have engaged with my comments and made improvements accordingly. Since my comments were not critical of the main results but rather or their presentation and the presentation has now improved, I am now happy with the submission and recommend publication in its current form.

  • validity: high
  • significance: high
  • originality: good
  • clarity: good
  • formatting: good
  • grammar: good

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