Large-space and large-time asymptotics for the mKdV soliton gas with any odd genus

Dedi Yan; Kedong Wang; Xianguo Geng

arxiv: 2605.19934 · v1 · pith:JI4F72H6new · submitted 2026-05-19 · 🌊 nlin.SI

Large-space and large-time asymptotics for the mKdV soliton gas with any odd genus

Dedi Yan , Xianguo Geng , Kedong Wang This is my paper

Pith reviewed 2026-05-20 03:43 UTC · model grok-4.3

classification 🌊 nlin.SI

keywords mKdV soliton gasodd genus asymptoticsRiemann theta functionlarge-time behaviornonlinear steepest descentg-functionRiemann-Hilbert problemuniform error estimates

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The pith

The mKdV soliton gas of genus 2n-1 has large positive-x asymptotics given by a single Riemann theta function while large-time behavior partitions the half-plane into 2n+1 regions each carrying its own theta-function description.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that an mKdV soliton gas built on an exact genus-2n-1 spectral curve possesses an explicit large-space limit: as x tends to positive infinity the solution is described by the Riemann theta function of that same genus. For large positive time the (x,t) half-plane is partitioned into 2n+1 distinct regions whose boundaries are determined by a g-function analysis. Inside each region the leading term is again a Riemann theta function, now of possibly lower genus, together with a uniform error bound obtained from contour deformation of the underlying Riemann-Hilbert problem. The construction works for every positive integer n and therefore covers every odd genus. These formulas give a concrete, computable prediction for the long-distance and long-time shape of the gas without repeated numerical solution of the nonlinear PDE.

Core claim

For the modified Korteweg-de Vries equation the soliton gas of exact genus 2n-1 admits the following asymptotic description: when x tends to positive infinity the solution converges to the Riemann-theta function associated with the genus-2n-1 curve; when t tends to positive infinity the half-plane is divided into 2n+1 regions separated by curves fixed by the g-function, and in each region the solution is asymptotically a Riemann-theta function of appropriate genus with a uniform error estimate derived from the nonlinear steepest descent method applied to the Riemann-Hilbert problem.

What carries the argument

Riemann-theta functions of genus 2n-1 (and lower) that arise after the g-function deforms the contour in the Riemann-Hilbert problem for the mKdV soliton gas.

If this is right

The same theta-function formulas hold uniformly throughout each of the 2n+1 temporal regions, including near their boundaries.
The partitioning into exactly 2n+1 regions is determined by the topology of the genus-2n-1 curve and remains fixed for all large t.
Error estimates remain uniform and do not deteriorate as the solution crosses from one region into another.
The result applies for every positive integer n and therefore describes soliton gases of every odd genus.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same region-counting and theta-function structure may persist when the spectral data are perturbed slightly away from exact genus 2n-1.
Numerical evolution of finite-genus mKdV solutions could be used to locate the predicted region boundaries and to measure the actual decay of the error term.
The method supplies a template that could be tried on other integrable equations whose Riemann-Hilbert problems admit g-function analysis.

Load-bearing premise

The initial data must be an exact-genus-2n-1 soliton gas whose associated Riemann-Hilbert problem permits a standard nonlinear steepest descent analysis with a g-function that has no additional restrictions on the spectral measure or branch points.

What would settle it

Numerical solution of the mKdV equation starting from a genus-1 or genus-3 soliton-gas datum, then direct comparison of the computed field at large positive x or inside one of the predicted large-t regions against the explicit Riemann-theta expression, checking whether the difference decays at the claimed rate.

Figures

Figures reproduced from arXiv: 2605.19934 by Dedi Yan, Kedong Wang, Xianguo Geng.

**Figure 2.** Figure 2: The homology basis for the Riemann surface [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** Figure 3: The jump contours C˜ j , C˜−j , C˜m,βm and C˜−m,βm. 38 [PITH_FULL_IMAGE:figures/full_fig_p038_3.png] view at source ↗

**Figure 4.** Figure 4: The homology basis for the Riemann surface [PITH_FULL_IMAGE:figures/full_fig_p041_4.png] view at source ↗

read the original abstract

We study the large-space and large-time asymptotic behavior of the soliton gas of genus $2n-1$ for the mKdV equation with $n\in \mathbb{N}_+$. As $x \to +\infty$, we show that the large-space asymptotics of the mKdV soliton gas can be expressed with the Riemann-theta function of genus $2n-1$. For large $t$, based on the nonlinear steepest descent method and $g$-function approach, we establish a global large-time asymptotic description of the mKdV soliton gas. The half-plane $\{(x,t):-\infty<x<+\infty, t>0\}$ is divided into $2n+1$ separated regions. In each region, the large-time asymptotics of the mKdV soliton gas is given by using the Riemann-theta functions and uniform error estimation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a uniform large-time and large-space asymptotic picture for mKdV soliton gases at arbitrary odd genus using standard RH steepest descent.

read the letter

The main point is that this work extends the asymptotic analysis of the mKdV soliton gas from specific low-genus cases to any odd genus 2n-1. As x goes to positive infinity the solution is expressed via the Riemann-theta function of genus 2n-1. For large t they divide the half-plane into 2n+1 regions and give Riemann-theta leading terms plus uniform error estimates in each region. They reach this by setting up the Riemann-Hilbert problem for the soliton gas and running the nonlinear steepest descent method with a g-function to handle the contour deformation globally. That uniform treatment across all odd genera is the actual new piece; earlier papers handled genus 1 or 3 separately and this organizes the whole family in one go. The approach follows the usual toolkit for these problems, so the formal steps line up with what has worked before. The abstract is clear about the region division and the error claims. One soft spot worth checking is the g-function step. The variational problem that produces g has to deliver a function with the strict inequality off the cuts for the deformation to work and for the 2n+1 regions to separate cleanly. If the branch points sit in non-generic positions or the density of states has accumulation points, that inequality can fail or turn into equality on some arcs. The paper states the result for any odd genus without listing extra conditions on the spectral data, so this is the place where the argument could need more restrictions or case handling. The rest of the derivation looks standard and the citation pattern is the expected one for steepest descent on soliton gases. This is for people who work on integrable nonlinear waves and soliton gases. A reader who wants a single framework for the different regimes in these gases will get something concrete from it. It deserves a serious referee because the claim is specific enough to verify and the methods are reproducible in principle.

Referee Report

1 major / 2 minor

Summary. The paper derives large-space asymptotics for the mKdV soliton gas of exact genus 2n-1, showing that as x → +∞ the solution is expressed via the Riemann-theta function of genus 2n-1. For large t > 0 it applies the nonlinear steepest-descent method together with a g-function to the associated Riemann-Hilbert problem, partitioning the half-plane into 2n+1 regions in each of which the asymptotics are given by a Riemann-theta function (of appropriate genus) together with a uniform error estimate.

Significance. If the derivations hold, the result supplies a global large-time asymptotic picture for soliton gases of arbitrary odd genus, extending earlier genus-1 and genus-3 analyses. The explicit use of the g-function and contour deformation to obtain uniform error bounds is a technical strength; the claim is falsifiable by direct numerical comparison in each of the 2n+1 regions.

major comments (1)

[Large-time asymptotics (g-function construction)] The global division of the half-plane into 2n+1 regions with Riemann-theta asymptotics (stated in the abstract and presumably proved in the large-t section) presupposes that the variational problem for the g-function admits a solution whose real part satisfies the strict inequality off the cuts for arbitrary positive density of states and arbitrary positions of the 2n branch points. The manuscript does not supply an explicit argument or genericity condition guaranteeing that the g-function exists and the inequality remains strict; if the spectral measure has accumulation points or the branch points are specially positioned, the construction may degenerate and the claimed partition into 2n+1 regions may fail. This is load-bearing for the central large-time claim.

minor comments (2)

Clarify the precise genus of the theta function used in each of the 2n+1 regions and state the error term (e.g., O(t^{-1/2})) uniformly in the region.
Add a short remark on how the large-space (x → +∞) result is recovered as a special case of the large-t analysis or whether it is proved independently.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comment below and will incorporate clarifications in the revised version.

read point-by-point responses

Referee: [Large-time asymptotics (g-function construction)] The global division of the half-plane into 2n+1 regions with Riemann-theta asymptotics (stated in the abstract and presumably proved in the large-t section) presupposes that the variational problem for the g-function admits a solution whose real part satisfies the strict inequality off the cuts for arbitrary positive density of states and arbitrary positions of the 2n branch points. The manuscript does not supply an explicit argument or genericity condition guaranteeing that the g-function exists and the inequality remains strict; if the spectral measure has accumulation points or the branch points are specially positioned, the construction may degenerate and the claimed partition into 2n+1 regions may fail. This is load-bearing for the central large-time claim.

Authors: We thank the referee for highlighting this key technical point. The g-function is constructed by solving the standard variational problem for the equilibrium measure associated with the logarithmic potential and the external field induced by the 2n branch points. Under the assumptions that the density of states is positive and continuous with no accumulation points on the real line, and that the branch points are distinct and in general position, the variational problem admits a unique solution satisfying the strict inequality Re(g) < 0 off the cuts. This is consistent with the theory of equilibrium measures in the presence of external fields and ensures the contour deformations yield the claimed 2n+1 regions with uniform error bounds. We agree that the manuscript would be strengthened by an explicit statement of these genericity conditions. In the revised version we will add a dedicated remark in the large-time section stating the assumptions on the density and branch points under which the g-function exists with the required properties, and briefly note that degenerate cases (accumulation points or specially positioned branch points) may lead to a reduced number of regions but lie outside the generic setting considered here. This revision clarifies the scope without altering the main results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies established steepest-descent techniques

full rationale

The paper derives large-space and large-time asymptotics for the mKdV soliton gas of genus 2n-1 by applying the nonlinear steepest descent method and g-function approach to the associated Riemann-Hilbert problem. These techniques are standard in integrable systems literature and the resulting division of the half-plane into 2n+1 regions with Riemann-theta function expressions and uniform error bounds follows directly from contour deformation and variational analysis of the g-function without reducing to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The claims for arbitrary odd genus rest on the general applicability of the method to the given spectral data and do not exhibit any step where the output is equivalent to the input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard tools of integrable systems without introducing new free parameters or postulated entities in the abstract.

axioms (2)

standard math Riemann theta functions of genus 2n-1 are well-defined and satisfy the required periodicity and analytic properties
Invoked for the large-space and large-time expressions.
domain assumption The nonlinear steepest descent method and g-function approach apply directly to the Riemann-Hilbert problem of the mKdV soliton gas
Central to obtaining the regional large-time asymptotics.

pith-pipeline@v0.9.0 · 5686 in / 1272 out tokens · 39022 ms · 2026-05-20T03:43:01.026761+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We establish a global large-time asymptotic description... divided into 2n+1 separated regions... Riemann-theta functions and uniform error estimation.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages · 1 internal anchor

[1]

M. A. Alejo, C. Mu˜ noz, Nonlinear stability of MKdV breathers, Comm. Math. Phys. 324 (2013) 233–262

work page 2013
[2]

Bertola, T

M. Bertola, T. Grava, G. Orsatti, Soliton shielding of the focusing nonlinear Schr¨ odinger equation, Phys. Rev. Lett. 130 (2023) 127201

work page 2023
[3]

Colliander, M

J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Sharp global well-posedness for KdV and modified KdV on R and T, J. Amer. Math. Soc. 16 (2003) 705–749

work page 2003
[4]

G. Chen, J. Q. Liu, Soliton resolution for the focusing modified KdV equation, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire 38 (2021) 2005–2071

work page 2021
[5]

G. Chen, J. Q. Liu, Long-time asymptotics to the modified KdV equation in weighted Sobolev spaces, Forum Math. Sigma 10 (2022) e66 1–52

work page 2022
[6]

Deift, A

P. Deift, A. Its, X. Zhou, A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics, Ann. of Math. 146 (1997) 149-235

work page 1997
[7]

Deift, X

P. Deift, X. Zhou, A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. 137 (1993) 295–368

work page 1993
[8]

Dyachenko, D

S. Dyachenko, D. Zakharov, V. Zakharov, Primitive potentials and bounded solutions of the KdV equation, Physica D 333 (2016) 148–156

work page 2016
[9]

G. A. El, The thermodynamic limit of the Whitham equations, Phys. Lett. A 311 (2003) 374–383

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[10]

G. A. El, A. M. Kamchatnov, Kinetic equation for a dense soliton gas, Phys. Rev. Lett. 95 (2005) 204101

work page 2005
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G. A. El, A. M. Kamchatnov, M. V. Pavlov, S. A. Zykov, Kinetic equation for a soliton gas and its hydrodynamic reductions, J. Nonlinear Sci. 21 (2010) 151–191

work page 2010
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G. A. El, A. Tovbis, Spectral theory of soliton and breather gases for the focusing nonlinear Schr¨ odinger equation, Phys. Rev. E 21 (2020) 052207. 47

work page 2020
[13]

A. A. Gelash, D. S. Agafontsev, Strongly interacting soliton gas and formation of rogue waves, Phys. Rev. E 98 (2018) 042210

work page 2018
[14]

A. A. Gelash, D. S. Agafontsev, V. E. Zakharov, G. A. El, S. Randoux, P. Suret, Bound state soliton gas dynamics underlying the spontaneous modulational instability, Phys. Rev. Lett. 123 (2019) 234102

work page 2019
[15]

Germain, F

P. Germain, F. Pusateri, F. Rousset, Asymptotic stability of solitons for mKdV equation, Adv. Math. 299 (2016) 272–330

work page 2016
[16]

Girotti, T

M. Girotti, T. Grava, R. Jenkins, K. T. R. McLaughlin, Rigorous asymptotics of a KdV soliton gas, Commun. Math. Phys. 384 (2021) 733–784

work page 2021
[17]

Girotti, T

M. Girotti, T. Grava, R. Jenkins, K. T. R. McLaughlin, A. Minakov, Soliton versus the gas: Fredholm determinants, analysis, and the rapid oscillations behind the kinetic equation, Commun. Pure Appl. Math. 76 (2023) 3233–3299

work page 2023
[18]

Grava, A

T. Grava, A. Minakov, On the long-time asymptotic behavior of the modified Korteweg–de Vries equation with step-like initial data, SIAM J. Math. Anal. 52 (2020) 5892–5993

work page 2020
[19]

X. F. Han, X. E. Zhang, H. H. Dong, Large x asymptotics of the soliton gas for the nonlinear Schr¨ odinger equation, Stud. Appl. Math. 154 (2025) e70027

work page 2025
[20]

Kenig, G

C. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm.Pure Appl. Math. 46 (1993) 527–620

work page 1993
[21]

Kotlyarov, D

V. Kotlyarov, D. Shepelsky, Planar unimodular Baker-Akhiezer function for the nonlinear Schr¨ odinger equation, Ann. Math. Sci. Appl. 2 (2017) 343–384

work page 2017
[22]

L. M. Ling, X. Sun, The multi elliptic-localized solutions and their asymptotic behaviors for the mKdV equation, Stud. Appl. Math. 150 (2023) 135–183

work page 2023
[23]

Suret, S

P. Suret, S. Randoux, A. Gelash, D. Agafontsev, B. Doyon, G. El, Soliton gas: Theory, numerics, and experiments, Phys. Rev. E 109 (2024) 061001

work page 2024
[24]

D. D. Yan, X. G. Geng, R. M. Li, M. X. Jia, Large-space and large-time asymptotics of the complex mKdV soliton gas with any even genus,submitted

work page
[25]

Tovbis, F.D

A. Tovbis, F.D. Wang, Recent developments in spectral theory of the focusing NLS soli- ton and breather gases: the thermodynamic limit of average densities, fluxes and certain meromorphic differentials; periodic gases, J. Phys. A 55 (2022) 424006

work page 2022
[26]

D. S. Wang, D. H. Zhu, X. D. Zhu, Genus two KdV soliton gases and their long-time asymptotics.arXiv:2410.22634. 48

work page internal anchor Pith review Pith/arXiv arXiv
[27]

Zakharov, Kinetic equation for solitons, Sov

V. Zakharov, Kinetic equation for solitons, Sov. Phys. JETP. 33 (1971) 538–541

work page 1971
[28]

G. Q. Zhang, Z. Y. Yan, Focusing and defocusing mKdV equations with nonzero boundary conditions: Inverse scattering transforms and soliton interactions, Physica D 410 (2020) 132521. 49

work page 2020

[1] [1]

M. A. Alejo, C. Mu˜ noz, Nonlinear stability of MKdV breathers, Comm. Math. Phys. 324 (2013) 233–262

work page 2013

[2] [2]

Bertola, T

M. Bertola, T. Grava, G. Orsatti, Soliton shielding of the focusing nonlinear Schr¨ odinger equation, Phys. Rev. Lett. 130 (2023) 127201

work page 2023

[3] [3]

Colliander, M

J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Sharp global well-posedness for KdV and modified KdV on R and T, J. Amer. Math. Soc. 16 (2003) 705–749

work page 2003

[4] [4]

G. Chen, J. Q. Liu, Soliton resolution for the focusing modified KdV equation, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire 38 (2021) 2005–2071

work page 2021

[5] [5]

G. Chen, J. Q. Liu, Long-time asymptotics to the modified KdV equation in weighted Sobolev spaces, Forum Math. Sigma 10 (2022) e66 1–52

work page 2022

[6] [6]

Deift, A

P. Deift, A. Its, X. Zhou, A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics, Ann. of Math. 146 (1997) 149-235

work page 1997

[7] [7]

Deift, X

P. Deift, X. Zhou, A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. 137 (1993) 295–368

work page 1993

[8] [8]

Dyachenko, D

S. Dyachenko, D. Zakharov, V. Zakharov, Primitive potentials and bounded solutions of the KdV equation, Physica D 333 (2016) 148–156

work page 2016

[9] [9]

G. A. El, The thermodynamic limit of the Whitham equations, Phys. Lett. A 311 (2003) 374–383

work page 2003

[10] [10]

G. A. El, A. M. Kamchatnov, Kinetic equation for a dense soliton gas, Phys. Rev. Lett. 95 (2005) 204101

work page 2005

[11] [11]

G. A. El, A. M. Kamchatnov, M. V. Pavlov, S. A. Zykov, Kinetic equation for a soliton gas and its hydrodynamic reductions, J. Nonlinear Sci. 21 (2010) 151–191

work page 2010

[12] [12]

G. A. El, A. Tovbis, Spectral theory of soliton and breather gases for the focusing nonlinear Schr¨ odinger equation, Phys. Rev. E 21 (2020) 052207. 47

work page 2020

[13] [13]

A. A. Gelash, D. S. Agafontsev, Strongly interacting soliton gas and formation of rogue waves, Phys. Rev. E 98 (2018) 042210

work page 2018

[14] [14]

A. A. Gelash, D. S. Agafontsev, V. E. Zakharov, G. A. El, S. Randoux, P. Suret, Bound state soliton gas dynamics underlying the spontaneous modulational instability, Phys. Rev. Lett. 123 (2019) 234102

work page 2019

[15] [15]

Germain, F

P. Germain, F. Pusateri, F. Rousset, Asymptotic stability of solitons for mKdV equation, Adv. Math. 299 (2016) 272–330

work page 2016

[16] [16]

Girotti, T

M. Girotti, T. Grava, R. Jenkins, K. T. R. McLaughlin, Rigorous asymptotics of a KdV soliton gas, Commun. Math. Phys. 384 (2021) 733–784

work page 2021

[17] [17]

Girotti, T

M. Girotti, T. Grava, R. Jenkins, K. T. R. McLaughlin, A. Minakov, Soliton versus the gas: Fredholm determinants, analysis, and the rapid oscillations behind the kinetic equation, Commun. Pure Appl. Math. 76 (2023) 3233–3299

work page 2023

[18] [18]

Grava, A

T. Grava, A. Minakov, On the long-time asymptotic behavior of the modified Korteweg–de Vries equation with step-like initial data, SIAM J. Math. Anal. 52 (2020) 5892–5993

work page 2020

[19] [19]

X. F. Han, X. E. Zhang, H. H. Dong, Large x asymptotics of the soliton gas for the nonlinear Schr¨ odinger equation, Stud. Appl. Math. 154 (2025) e70027

work page 2025

[20] [20]

Kenig, G

C. Kenig, G. Ponce, L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm.Pure Appl. Math. 46 (1993) 527–620

work page 1993

[21] [21]

Kotlyarov, D

V. Kotlyarov, D. Shepelsky, Planar unimodular Baker-Akhiezer function for the nonlinear Schr¨ odinger equation, Ann. Math. Sci. Appl. 2 (2017) 343–384

work page 2017

[22] [22]

L. M. Ling, X. Sun, The multi elliptic-localized solutions and their asymptotic behaviors for the mKdV equation, Stud. Appl. Math. 150 (2023) 135–183

work page 2023

[23] [23]

Suret, S

P. Suret, S. Randoux, A. Gelash, D. Agafontsev, B. Doyon, G. El, Soliton gas: Theory, numerics, and experiments, Phys. Rev. E 109 (2024) 061001

work page 2024

[24] [24]

D. D. Yan, X. G. Geng, R. M. Li, M. X. Jia, Large-space and large-time asymptotics of the complex mKdV soliton gas with any even genus,submitted

work page

[25] [25]

Tovbis, F.D

A. Tovbis, F.D. Wang, Recent developments in spectral theory of the focusing NLS soli- ton and breather gases: the thermodynamic limit of average densities, fluxes and certain meromorphic differentials; periodic gases, J. Phys. A 55 (2022) 424006

work page 2022

[26] [26]

D. S. Wang, D. H. Zhu, X. D. Zhu, Genus two KdV soliton gases and their long-time asymptotics.arXiv:2410.22634. 48

work page internal anchor Pith review Pith/arXiv arXiv

[27] [27]

Zakharov, Kinetic equation for solitons, Sov

V. Zakharov, Kinetic equation for solitons, Sov. Phys. JETP. 33 (1971) 538–541

work page 1971

[28] [28]

G. Q. Zhang, Z. Y. Yan, Focusing and defocusing mKdV equations with nonzero boundary conditions: Inverse scattering transforms and soliton interactions, Physica D 410 (2020) 132521. 49

work page 2020