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Traveling periodic waves and breathers in the nonlocal derivative NLS equation

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Pith's one-line read This paper proves nonlinear stability of the constant background in the defocusing nonlocal derivative NLS equation and constructs exact $N$-breather solutions on traveling periodic wave backgrounds in closed determinant form.

desk verdict Solid stability theorems plus plausible but under-verified N-breather formulas; the boundedness argument for the negative-speed branch cites a reference that doesn't cover it. read the letter →

arxiv 2501.15625 v1 pith:6QHGI4MJ submitted 2025-01-26 nlin.SI math-phmath.APmath.DSmath.MPnlin.PS

Jinbing Chen , Dmitry E. Pelinovsky This is my paper · ORCID
classification nlin.SImath-phmath.APmath.DSmath.MPnlin.PS MSC 35Q5537K1537K4035B35
keywords nonlocalderivativeNLSequationHirotabilinearmethodtravelingperiodicwavesbreathersLaxspectrumnonlinearstabilitydeterminantsolutionsenvelope
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies the nonlocal derivative nonlinear Schrödinger equation, a model for envelope-wave modulations in stratified fluids that also arises as a continuum limit of Calogero–Moser–Sutherland particle dynamics. Its main results are twofold: the nonzero constant background $u=1$ is stable — linearly for decaying perturbations in the defocusing equation and nonlinearly in $H^1_{\rm per}$ for every period, with the focusing case holding only under restrictions — and every family of traveling periodic waves constructed by Hirota's bilinear method carries explicit breather solutions. The general $N$-breather solution is written as a quotient of determinants, one formula on the nonzero background and one on the zero background. If these results are correct, the two versions of the equation do not behave like the cubic NLS 'defocusing'/'focusing' pair: the constant background does not exhibit rogue waves, and solitary waves propagate steadily on periodic backgrounds in both versions.

What carries the argument

The machinery is Hirota's bilinear method combined with the Lax-pair representation of the equation. Writing $u=g/f$ and $|u|^2=1-i\sigma\partial_x\ln(f/\tilde f)$ turns the NDNLS equation and its linear system into bilinear equations whose elementary exponential solutions come from the Benjamin–Ono hierarchy. The load-bearing step is degeneration: taking the long-wave limit $k_j\to 0$ in the $(N+1)$-periodic determinant solutions yields the breather determinants (4.45) and (5.25), with the admissible ranges of the solitary-wave speed $c_2$ read off from the Lax spectrum, $\Sigma=[\lambda_0,\lambda_0+k_1]\cup[\sigma,\infty)$ on the nonzero background and $\Sigma=[0,k_1]\cup[0,\infty)$ on the zero background. Zero-location lemmas inherited from the Benjamin–Ono theory keep the zeros of $f$ and $\tilde f$ in opposite half-planes, which is what makes the quotient bounded.

What would settle it

Choose the $N=2$ breather from formula (4.45) with generic admissible parameters, substitute it into the NDNLS equation (2.1) with high-precision numerics, and test whether the residual converges to zero as the grid resolves the periodic background; separately, for the second family in Theorem 4, compute the complex zeros of $f$ for $c_2\in(-\infty,-|k_2|-2k_1)$ across a range of $k_1,k_2$ and check whether any zero reaches the real axis, which would make the solution unbounded.

Watch

Extended reading notes

Core claim

The central claim is that the apparent 'defocusing'/'focusing' dichotomy of the nonlocal derivative NLS equation is not reflected in the stability of the constant background or in the existence of exact solitary waves. For the defocusing version, Theorem 2 establishes that $u=1$ is nonlinearly stable in $H^1_{\rm per}(0,L)$ for every period $L>0$: sufficiently small perturbations stay close to the background up to a phase rotation for all time. For the focusing version the same Lyapunov argument works only when $L<\pi$, and linear stability holds only for non-resonant decaying data with $\hat v_0(\pm1)=0$. On the constructive side, the paper obtains traveling periodic waves in elementary trigonometric form, computes their Lax spectrum, and takes long-wave limits of inherited multi-periodic solutions to obtain single breathers and, in closed determinant form, the general $N$-breather solution: $u=e^{(\phi_1-\psi_1)/2}\det\hat G/\det\hat F$ on the nonzero background and $u=\gamma_1\det\hat G/\det\hat F$ on the zero background.

Load-bearing premise

The paper's new breathers are obtained by taking long-wave limits of multi-periodic solutions taken from earlier work, whose correctness and zero-location properties the paper relies on rather than re-derives; if any inherited formula or zero-location lemma failed, or if the limiting procedure exchanged limits invalidly, the determinant quotient would not solve the NDNLS equation.

Editorial extensions

If this is right

  • Small periodic perturbations of the defocusing background $u=1$ never grow: the $H^1_{\rm per}$ distance to the background, up to a phase rotation, is controlled for all time.
  • The breather formulas describe dark solitary waves on the periodic background in the defocusing case and both bright and dark waves in the focusing case; in the nonzero-background families the phase shift after the soliton passes equals one period of the periodic wave.
  • The $N$-breather determinants reduce to the single-breather solutions when $N=1$, and to the algebraic (long-wave) solitons as $k_1\to 0$, so the paper's construction covers both single and multi-soliton states in one uniform expression.
  • Breathers associated with isolated eigenvalues of the Lax spectrum and those associated with embedded eigenvalues show the same qualitative dynamics, even though the spectral situations differ.
  • The labels 'defocusing' and 'focusing' are not dynamically justified for this equation: both versions have stable constant backgrounds in suitable settings and support solitary waves traveling on periodic backgrounds.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A possible extension: because the construction only uses long-wave degeneration of multi-periodic determinants, the same scheme should produce explicit breathers on periodic backgrounds for the intermediate NLS equation interpolating between the shallow- and deep-fluid limits.
  • A natural test: since nonlinear stability is proven for $L<\pi$ while a linear resonance appears at period $2\pi$, tracking perturbations across $L=\pi$ numerically would reveal whether the stability threshold is sharp.
  • A robustness check: the zero-location proof for the second breather family uses a modulus-contradiction argument; computing the complex zeros of $f$ numerically across the full parameter range would confirm the no-crossing conclusion at finite precision.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript studies the nonlocal derivative NLS equation (2.1) in both defocusing and focusing versions. It proves linear stability of the nonzero constant background for decaying perturbations (Theorem 1) and nonlinear stability for periodic perturbations in the defocusing case for every period L>0, and in the focusing case for L<π (Theorem 2 and Corollary 1). It then constructs traveling periodic waves, computes their Lax spectra (Propositions 1–4), derives single-breather solutions on these backgrounds (Theorem 3 and Corollary 2), and presents N-breather solutions in closed determinant form, Eqs. (4.45) and (5.25). The stability proofs are largely self-contained; the exact solutions use the Hirota bilinear method and long-wave limits of multi-periodic solutions quoted from earlier work of Matsuno and from Dobrokhotov–Krichever.

Significance. If the main claims hold, the paper gives a valuable picture of an integrable nonlocal model: stable constant backgrounds, absence of rogue-wave behavior in the considered classes, a complete classification of periodic waves with their Lax spectra, and explicit breather solutions on periodic backgrounds. The Fourier proof of Theorem 1 is explicit, and the N=1 breather constructions in Theorem 3 and Corollary 2 are verified in detail, including zero-location arguments. The paper is also honest about several limitations, such as the open problem for focusing L≥π. However, the general N-breather determinant formulas are not verified independently in the text and inherit validity from quoted multi-periodic solutions; one of the boundedness claims (Remark 17) is not supported for all parameter families. These issues affect the exact-solution part of the paper's central claims and require repair.

major comments (3)
  1. [§5.4 and Remark 17] The general N-breather formula (5.25) is asserted for two families of wave speeds, c_j∈(0,∞) and c_j∈(-∞,-2k_1). However, the underlying (N+1)-periodic solution is quoted from [41], and Remark 15 explicitly states that [41] covers only the family c_2∈(|k_2|,∞) and misses c_2∈(-∞,-|k_2|-2k_1). Consequently, the appeal in Remark 17 to [41, Proposition 2] for boundedness of the N-breathers is not valid for the second family when N>1. No substitute zero-location proof or bilinear substitution is given for the determinant matrices (5.23)–(5.24) in that family. The N=1 case in Corollary 2 is proved, but the general N>1 formula for the second family is not established.
  2. [§4.4 and Remark 9] The N-breather formula (4.45) is obtained by taking the long-wave limits k_j→0 for 2≤j≤N+1 in (N+1)-periodic solutions quoted from [37]. The matrix entries (4.41)–(4.44) are stated without derivation, and no verification that the determinant quotient solves the bilinear system (4.3) is shown. The boundedness argument in Remark 9 rests on a continuity claim that the zeros of f and \tilde f cannot cross the real line in the limit, but for N>1 this is not proved; the pre-limit zero-location is quoted from [18, Lemma 1.1], and the limiting determinant's zero-location is asserted. The N=1 case is handled in Theorem 3, but the general N-breather claim is conditional on unshown algebraic limits and zero-location statements.
  3. [§3, Eq. (3.9)] The control of the mean value of the real part of v in the proof of Theorem 2 contains an invalid inequality for negative means. The inequality |\hat v_0(t)| ≤ (√(L+I_1)/√L) − 1 fails when I_1<0, because the right-hand side is negative; this is exactly the regime of small negative means. The argument needs a corrected branch-selection step: one should use continuity of \hat v_0(t) to stay on the root \hat v_0 = −1 + √(1 + I_1/L − ∑_{n≠0}|\hat v_n|^2) and then bound |\hat v_0| by the smallness of I_1 and the spatial variation. This is a load-bearing step for nonlinear stability, though it appears fixable without changing the result.
minor comments (5)
  1. [§4.3 before Theorem 3] The sentence 'This result of [18] holds for the nonlocal model (2.1) because the functional representations of f and \tilde f in (4.25) is identical to that for the BO equation' would be more convincing if it named the precise structural property of the pair (f,\tilde f) that transfers the lemma from the BO equation to this model.
  2. [§4.4] After Eq. (4.44) the text says that \hat{\tilde F} and \hat{\tilde G} are related to the complex-conjugate versions of \hat F and \hat G, but the precise relation is not displayed; please give the explicit connection, since the formula for |u|^2 in (4.45) uses \det \bar{\hat F}.
  3. [§5.3, Corollary 2] The zero-location proof is written for the zeros of f only. A sentence stating that the same argument applies verbatim to \tilde f (or a short display of the analogous equation) would remove any ambiguity, since the boundedness of the solution needs both.
  4. [§5.4] The text says the (N+1)-periodic solution is taken from [41] but extended 'in a more general setting as in Theorem 4.' This extension is not automatic for N>1; please state precisely which parameter intervals are being extended and how the zero-location for the extended families is obtained for all N.
  5. [Throughout] There are a few typographical errors, e.g., 'travelng' in §4.1, 'propagatng' in §5.3, 'caes' in §5.3, and 'doe not' in §5.3. These should be corrected in a final revision.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: central claims reduce to limits of quoted external multi-periodic solutions and to conserved quantities from prior work, not to their own outputs; the one self-citation is non-load-bearing.

full rationale

The stability results (Theorems 1-2, Corollary 1) are self-contained deductions from the linearized equation and from conserved quantities I1, I2, I3 quoted from [45] and [41]; the Lyapunov functional (3.7) is expanded directly in v and shown coercive, so no fitted parameter or predicted quantity is being reused. The traveling periodic waves (Propositions 1 and 3) are derived in the text from the bilinear equations. The single breathers (Theorem 3, Corollary 2) are obtained as long-wave limits k2 -> 0 of the 2-periodic solutions (4.25)/(5.19), which are quoted from Matsuno [37,41] and checked by substitution; zero locations are justified from [18, Lemma 1.1] and by explicit modulus-contradiction arguments, not by assuming the target solutions. The N-breather determinants (4.45) and (5.25) are formally limits of the (N+1)-periodic determinant solutions of [37,41]; this is a derivation-chain dependence on external formulas rather than circularity. The only self-citation, [14], is used in Remark 8 only as a proof-technique comparison and is explicitly corrected; it does not carry the central claims. Recognized gaps - the unshown substitution verification in Theorem 4, the lack of a fully displayed convergence proof for the determinant limits, and the questionable extension of [41, Prop. 2] to the second parameter family in Remark 17 - are correctness risks, not circular reductions.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper contributes no fitted free parameters and no new physical entities. It relies on four external pillars: local well-posedness, Matsuno's multi-periodic solutions, the zero-location lemma, and the conserved quantities. These are standard and cited; the circularity burden is low.

assumptions (4)
  • domain assumption Local well-posedness of the NDNLS equation in H^1_per(R/LZ) exists for all time for small initial data.
    Invoked in Theorem 2 to extend local solutions globally; cited to [3,42], not proven in this paper.
  • domain assumption The explicit (N+1)-periodic wave solutions of Matsuno ([37,41]) satisfy the bilinear equations and have the stated parameter restrictions.
    Used as the starting point for the long-wave limits that produce breather solutions; the paper verifies some substitutions only by assertion.
  • domain assumption Zeros of f and tilde f for the periodic wave solutions stay in the lower and upper half-planes under the parameter restrictions, per [18, Lemma 1.1] and [41, Proposition 2].
    Guarantees boundedness of u = g/f and validity of the analytic projection formulas.
  • domain assumption The quantities I1, I2, I3 are conserved for the NDNLS equation on R and on the torus.
    Quoted from [45] and [41]; used to define the Lyapunov functional in Theorem 2.

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Pith. "Pith review of Traveling periodic waves and breathers in the nonlocal derivative NLS equation." pith.science (2026). https://pith.science/paper/6QHGI4MJ

@misc{pith2026250115625,
  author       = {Pith},
  title        = {Pith review of: Traveling periodic waves and breathers in the nonlocal derivative NLS equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6QHGI4MJ}},
  note         = {Machine review of arXiv:2501.15625}
}
abstract

A nonlocal derivative NLS (nonlinear Schr\"{o}dinger) equation describes modulations of waves in a stratified fluid and a continuous limit of the Calogero--Moser--Sutherland system of particles. For the defocusing version of this equation, we prove the linear stability of the nonzero constant background for decaying and periodic perturbations and the nonlinear stability for periodic perturbations. For the focusing version of this equation, we prove linear and nonlinear stability of the nonzero constant background under some restrictions. For both versions, we characterize the traveling periodic wave solutions by using Hirota's bilinear method, both on the nonzero and zero backgrounds. For each family of traveling periodic waves, we construct families of breathers which describe solitary waves moving across the stable background. A general breather solution with $N$ solitary waves propagating on the traveling periodic wave background is derived in a closed determinant form.

Figures

Figures reproduced from arXiv: 2501.15625 by the authors.

Figure 1
Figure 1. The profile of |u| 2 versus x for σ = +1, k1 = 0.25, and either c1 = −1 (left) or c1 = −0.5 (right) [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. The profile of |u| 2 versus x for σ = −1, k1 = 0.25, and either c1 = 2 + 2k1 (left) or c1 = 2 + 4k1 (right) [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. The Lax spectrum for the breather solutions of [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: The solution surface of |u| 2 for the breather versus (x + t, t) for k1 = 0.25, c1 = −1, and c2 = −0.5 [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: The solution surface of |u| 2 for the breather versus (x + t, t) for k1 = 0.25, c1 = −0.5, and c2 = −1. Proposition 2 and Remark 7. The first two cases in (4.33) are shown in [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]
Figure 6
Figure 6. Figure 6: The Lax spectrum for the breather solutions of [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]
Figure 7
Figure 7. Figure 7: The solution surface of |u| 2 for the breather versus (x − c1t, t) for k1 = 0.25, c1 = 2 + 2k1, and c2 = c1 + 2k1. relative to the periodic wave [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]
Figure 8
Figure 8. Figure 8: The solution surface of |u| 2 for the breather versus (x − c1t, t) for k1 = 0.25, c1 = 2 + 2k1, and c2 = c1 − 3 2 k1 [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]
Figure 9
Figure 9. Figure 9: The solution surface of |u| 2 for the breather versus (x − c1t, t) for k1 = 0.25, c1 = 2 + 2k1, and c2 = −k1. The Lax spectrum of the breather solution includes and additional eigenvalue − c2 2 relative to the Lax spectrum Σ = [−1,∞) of the traveling periodic wave. The…
Figure 10
Figure 10. Figure 10: The profile of |u| 2 versus x for σ = −1, k1 = 0.25, and either ϕ1 = 1 (left) or ϕ1 = 0.5 (right). 5.2. Lax spectrum of the traveling periodic wave. To obtain the exact solutions of the linear system (2.5) with σ = −1, we use the representations (4.13) and (5.1). The …
Figure 11
Figure 11. Figure 11: The solution surface of |u| 2 for the breather versus (x, t) for k1 = 0.25, ϕ1 = 1, c1 = −k1, and c2 = k1 [PITH_FULL_IMAGE:figures/full_fig_p034_11.png]
Figure 12
Figure 12. Figure 12: The solution surface of |u| 2 for the breather versus (x, t) for k1 = 0.25, ϕ1 = 1, c1 = −k1, and c2 = −3k1. Figures 11 and 12 display the solution surfaces (side view on the left and top view on the right) for the breather solution of Corollary 2 with two choices for…

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Works this paper leans on

52 extracted references · 47 canonical work pages

  1. [41]

    Multiphase solutions and their reductions for a nonlocal nonlinear Schr¨ odinger equation with focusing nonlinearity

    Y. Matsuno, “Multiphase solutions and their reductions for a nonlocal nonlinear Schr¨ odinger equation with focusing nonlinearity”, Stud. Appl. Math. 151 (2023) 883–922

    work page 2023

  2. [37]

    Multiperiodic and multisoliton solutions of a nonlocal nonlinear Schr¨ odinger equation for envelope waves

    Y. Matsuno, “Multiperiodic and multisoliton solutions of a nonlocal nonlinear Schr¨ odinger equation for envelope waves”, Phys. Lett. A 278 (2000) 53–58

    work page 2000

  3. [1]

    Integrable hydrodynamics of Calogero–Sutherland model: bidirectional Benjamin–Ono equation

    A. G. Abanov, E. Betterlheim, and P. B. Wiegmann, “Integrable hydrodynamics of Calogero–Sutherland model: bidirectional Benjamin–Ono equation”, J. Phys. A: Math. Theor. 42 (2009) 135201 (24 pages)

    work page 2009

  4. [2]

    Spectrally accurate fully discrete schemes for some non- local and nonlinear integrable PDEs via explicit formulas

    Y. Alama Bronsard, X. Chen, M. Dolbeault, “Spectrally accurate fully discrete schemes for some non- local and nonlinear integrable PDEs via explicit formulas”, arXiv: 2412.13480 (2024)

    work page arXiv 2024

  5. [3]

    On the global well-posedness of the Calogero-Sutherland derivative nonlinear Schr¨ odinger equation

    R. Badreddine, “On the global well-posedness of the Calogero-Sutherland derivative nonlinear Schr¨ odinger equation”, Pure Appl. Anal.6 (2024) 379–414

    work page 2024

  6. [4]

    Traveling waves & finite gap potentials for the Calogero-Sutherland Derivative nonlinear Schr\"odinger equation

    R. Badreddine, “Traveling waves and finite gap potentials for the Calogero–Sutherland derivative non- linear Schr¨ odinger equation”, Ann. Inst. H. Poincar´ e C Anal. Non Lineare arXiv: 2307.01592 (2023)

    work page Pith review arXiv 2023

  7. [5]

    Zero dispersive limit of the Calogero–Moser derivative NLS equation

    R. Badreddine, “Zero dispersive limit of the Calogero–Moser derivative NLS equation”, SIAM J. Math. Anal. 56 (2024) 7228–7249

    work page 2024

  8. [6]

    Spin generalizations of the Benjamin–Ono equation

    B. K. Berntson, E. Langmann, and J. Lenells, “Spin generalizations of the Benjamin–Ono equation”, Lett. Math. Phys. 112 (2022) 50 (45 pages)

    work page 2022

Show all 52 references
  1. [7]

    A focusing–defocusing intermediate nonlinear Schr¨ odinger system

    B. K. Berntson and A. Fagerlund, “A focusing–defocusing intermediate nonlinear Schr¨ odinger system”, Physica D 451 (2023) 133762 (26 pages)

    2023

  2. [8]

    Partial degeneration of finite gap solutions to the Korteweg–de Vries equation: soliton gas and scattering on elliptic background

    M. Bertola, R. Jenkins, and A. Tovbis, “Partial degeneration of finite gap solutions to the Korteweg–de Vries equation: soliton gas and scattering on elliptic background”, Nonlinearity 36 (2023) 3622–3660

    2023

  3. [9]

    Long-time asymptotics for the focusing nonlinear Schr¨ odinger equa- tion with nonzero boundary conditions at infinity and asymptotic stage of modulational instability

    G. Biondini and D. Mantzavinos, “Long-time asymptotics for the focusing nonlinear Schr¨ odinger equa- tion with nonzero boundary conditions at infinity and asymptotic stage of modulational instability”, Comm. Pure Appl. Math. 70 (2017) 2300–2365

    2017

  4. [10]

    Long-time asymptotics for the focusing nonlinear Schr¨ odinger equation with nonzero boundary conditions in the presence of a discrete spectrum

    G. Biondini, S. Li, and D. Mantzavinos, “Long-time asymptotics for the focusing nonlinear Schr¨ odinger equation with nonzero boundary conditions in the presence of a discrete spectrum”, Comm. Math. Phys. 382 (2021) 1495–1577

    2021

  5. [11]

    Rogue waves on the background of periodic standing waves in the derivative nonlinear Schr¨ odinger equation

    J. Chen and D. E. Pelinovsky, “Rogue waves on the background of periodic standing waves in the derivative nonlinear Schr¨ odinger equation”, Phys. Rev. E103 (2021) 062206 (25 pages)

    2021

  6. [12]

    Periodic waves in the discrete MKDV equation: modulational instability and rogue waves

    J. Chen and D. E. Pelinovsky, “Periodic waves in the discrete MKDV equation: modulational instability and rogue waves”, Physica D 445 (2023) 133652 (16 pages)

    2023

  7. [13]

    Rogue waves arising on the standing periodic waves in the Ablowitz- Ladik equation

    J. Chen and D. E. Pelinovsky, “Rogue waves arising on the standing periodic waves in the Ablowitz- Ladik equation”, Stud. Appl. Math. 152 (2024) 147–173

    2024

  8. [14]

    Bright and dark breathers of the Benjamin-Ono equation on the traveling periodic background

    J. Chen and D. E. Pelinovsky, “Bright and dark breathers of the Benjamin-Ono equation on the traveling periodic background”, Wave Motion 126 (2024) 103263 (10 pages)

    2024

  9. [15]

    Modulational instability of periodic standing waves in the derivative NLS equation

    J. Chen, D. E. Pelinovsky, and J. Upsal, “Modulational instability of periodic standing waves in the derivative NLS equation”, J. Nonlin. Sci. 31 (2021) 58 (32 pages)

    2021

  10. [16]

    Periodic standing waves in the focusing nonlinear Schr¨ odinger equation: Rogue waves and modulation instability

    J. Chen, D. E. Pelinovsky, and R. E. White, “Periodic standing waves in the focusing nonlinear Schr¨ odinger equation: Rogue waves and modulation instability”, Physica D 405 (2020) 132378 (13 pages)

    2020

  11. [17]

    The physics of the one-dimensional nonlinear Schr¨ odinger equation in fiber optics: Rogue waves, modulation instability and self-focusing phenomena

    F. Copie, S. Randoux, and P. Suret, “The physics of the one-dimensional nonlinear Schr¨ odinger equation in fiber optics: Rogue waves, modulation instability and self-focusing phenomena”, Rev. Phys. 5 (2020) 100037 (17 pages)

    2020

  12. [18]

    Multiphase solutions of the Benjamin–Ono equation and their averaging

    S. Yu. Dobrokhotov and I. M. Krichever, “Multiphase solutions of the Benjamin–Ono equation and their averaging”, Math. Notes 49 (1991) 583–594

    1991

  13. [19]

    Rogue waves and analogies in optics and oceanography

    J. M. Dudley, G. Genty, A. Mussot, A. Chabchoub, and F. Dias, “Rogue waves and analogies in optics and oceanography”, Nat. Rev. Phys. 1 (2019) 675–689 BREATHERS IN THE NONLOCAL DERIV ATIVE NLS EQUATION 39

    2019

  14. [20]

    Dispersive shock waves and modulation theory

    G. A. El and M. A. Hoefer, “Dispersive shock waves and modulation theory”, Physica D 133 (2016) 11–65

    2016

  15. [21]

    Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws

    G. A. El, M. A. Hoefer, and M. Shearer, “Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws”, SIAM Rev. 59 (2017) 3–61

    2017

  16. [22]

    Multi-breathers and high order rogue waves for the nonlinear Schr¨ odinger equation on the elliptic function background

    B. F. Feng, L. Ling, and D. A. Takahashi, “Multi-breathers and high order rogue waves for the nonlinear Schr¨ odinger equation on the elliptic function background”, Stud. Appl. Math.144 (2020) 46–101

    2020

  17. [23]

    Fibich, The Nonlinear Schr¨ odinger Equation: Singular Solutions and Optical Collapse(New York: Springer, 2015)

    G. Fibich, The Nonlinear Schr¨ odinger Equation: Singular Solutions and Optical Collapse(New York: Springer, 2015)

    2015

  18. [24]

    On the integrability of the Benjamin–Ono equation on the torus

    P. G´ erard and T. Kappeler, “On the integrability of the Benjamin–Ono equation on the torus”, Comm. Pure Appl. Math. 74 (2021) 1685–1747

    2021

  19. [25]

    On the spectrum of the Lax operator of the Benjamin–Ono equation on the torus

    P. G´ erard, T. Kappeler, and P. Topalov, “On the spectrum of the Lax operator of the Benjamin–Ono equation on the torus”, J. Funct. Anal. 279 (2020) 108762 (75 pages)

    2020

  20. [26]

    On the Benjamin–Ono equation on T and its periodic and quasiperiodic solutions

    P. G´ erard, T. Kappeler, and P. Topalov, “On the Benjamin–Ono equation on T and its periodic and quasiperiodic solutions”, J. Spectr. Theor. 12 (2022) 169–193

    2022

  21. [27]

    The Calogero–Sutherland derivative nonlinear Schr¨ odinger equation

    P. G´ erard and E. Lenzmann, “The Calogero–Sutherland derivative nonlinear Schr¨ odinger equation”, Comm. Pure Appl. Math. 77 (2024) 4008–4062

    2024

  22. [28]

    Rigorous asymptotics of a KdV soliton gas

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

    2021

  23. [29]

    Soliton versus the gas: Fredholm determinants, analysis, and the rapid oscillations behind the kinetic equation

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

    2023

  24. [30]

    KdV breathers on cnoidal wave background

    M. Hoefer, A. Mucalica, and D. E. Pelinovsky, “KdV breathers on cnoidal wave background”, J. Phys. A: Math. Theor. 56 (2023) 185701 (25 pages)

    2023

  25. [31]

    Turbulent threshold for continuum Calogero–Moser models

    J. Hogan and M. Kowalski, “Turbulent threshold for continuum Calogero–Moser models”, Pure Appl. Anal. 6 (2024) 941–954

    2024

  26. [32]

    P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-Gonz´ alez,The defocusing nonlinear Schr¨ odinger equation: From dark solitons to vortices and vortex rings, (Society for Industrial and Applied Mathe- matics, Philadelphia, PA, 2015)

    2015

  27. [33]

    Scaling-critical well-posedness for continuum Calogero–Moser models

    R. Killip, T. Laurens, and M. Vi¸ san, “Scaling-critical well-posedness for continuum Calogero–Moser models”, arXiv: 2311.12334 (2023)

    2023 arXiv

  28. [34]

    Construction of smooth chiral finite-time blow-up solutions to Calogero– Moser derivative nonlinear Schr¨ odinger equation

    K. Kim, T. Kim, and S. Kwon, “Construction of smooth chiral finite-time blow-up solutions to Calogero– Moser derivative nonlinear Schr¨ odinger equation”, arXiv:2404.09603 (2024)

    2024 arXiv

  29. [35]

    Soliton resolution for Calogero–Moser derivative nonlinear Schr¨ odinger equation

    T. Kim and S. Kwon, “Soliton resolution for Calogero–Moser derivative nonlinear Schr¨ odinger equation”, arXiv:2408.12843 (2024)

    2024

  30. [36]

    Multi-elliptic-dark soliton solutions in the defocusing nonlinear Schr¨ odinger equa- tion

    L. Ling and X. Sun, “Multi-elliptic-dark soliton solutions in the defocusing nonlinear Schr¨ odinger equa- tion”, Appl. Math. Lett. 148 (2023) 108866 (9 pages)

    2023

  31. [38]

    Linear stability of multiple dark solitary wave solutions of a nonlocal nonlinear Schr¨ odinger equation for envelope waves

    Y. Matsuno, “ Linear stability of multiple dark solitary wave solutions of a nonlocal nonlinear Schr¨ odinger equation for envelope waves”, Phys. Lett. A285 (2001) 286–292

    2001

  32. [39]

    Asymptotic solutions of the nonlocal nonlinear Schr¨ odinger equation in the limit of small dispersion

    Y. Matsuno, “Asymptotic solutions of the nonlocal nonlinear Schr¨ odinger equation in the limit of small dispersion”, Phys. Lett. A 309 (2003) 83–89

    2003

  33. [40]

    New representations of multiperiodic and multisoliton solutions for a class of nonlocal soliton equations

    Y. Matsuno, “New representations of multiperiodic and multisoliton solutions for a class of nonlocal soliton equations”, J. Phys. Soc. Jpn. 73 (2004) 3285–3293

    2004

  34. [42]

    Local well–posedness for the nonlocal nonlinear Schr¨ odinger equation below the energy space

    R. P. de Moura and D. Pilod, “Local well–posedness for the nonlocal nonlinear Schr¨ odinger equation below the energy space”, Adv. Diff. Eqs. 15 (2010) 925–952

    2010

  35. [43]

    Dark breathers on the snoidal wave background in the defocusing mKdV equation

    A. Mucalica and D. E. Pelinovsky, “Dark breathers on the snoidal wave background in the defocusing mKdV equation”, Lett. Math. Phys. 114 (2024) 100 (28 pages) 40 JINBING CHEN AND DMITRY E. PELINOVSKY

    2024

  36. [44]

    Intermediate nonlinear Schro¨ odinger equation for internal waves in a fluid of finite depth

    D. E. Pelinovsky, “Intermediate nonlinear Schro¨ odinger equation for internal waves in a fluid of finite depth”, Phys. Lett. A 197 (1995) 401–406

    1995

  37. [45]

    A spectral transform for the intermediate nonlinear Shr¨ odinger equation

    D. E. Pelinovsky and R. H. J. Grimshaw, “A spectral transform for the intermediate nonlinear Shr¨ odinger equation”, J. Math. Phys.36 (1995) 4203–4219

    1995

  38. [46]

    Nonlocal models for envelope waves in a stratified fluid

    D. E. Pelinovsky and R. H. J. Grimshaw, “Nonlocal models for envelope waves in a stratified fluid”, Stud. Appl. Math. 97 (1996) 369–391

    1996

  39. [47]

    The dark soliton on a cnoidal wave background

    H. Shin, “The dark soliton on a cnoidal wave background”, J. Phys. A: Math. Gen. 38 (2005) 3307–3315

    2005

  40. [48]

    Rogue waves in the sea: observations, physics and mathematics

    A. V. Slunyaev, D. E. Pelinovsky, and E. N. Pelinovsky, “Rogue waves in the sea: observations, physics and mathematics”, Phys. Usp. 66 (2023) 148–172

    2023

  41. [49]

    The intertwined derivative Schr¨ odinger system of Calogero–Moser–Sutherland type

    R. Sun, “The intertwined derivative Schr¨ odinger system of Calogero–Moser–Sutherland type”, Lett. Math. Phys. 114 (2024) 74 (32 pages)

    2024

  42. [50]

    D. A. Takahashi, “Integrable model for density-modulated quantum condensates: Solitons passing through a soliton lattice“, Phys. Rev. E 93 (2016) 062224 (20 pages)

    2016

  43. [51]

    Quantized blow up dynamics for derivative Calogero-Moser nonlinear Schr¨ odinger equation

    U. Jeong and T. Kim, “Quantized blow up dynamics for derivative Calogero-Moser nonlinear Schr¨ odinger equation”, arXiv:2412.12518 (2024)

    2024 arXiv

  44. [52]

    Freezing limits for Calogero–Moser–Sutherland particle models

    M. Voit, “Freezing limits for Calogero–Moser–Sutherland particle models”, Stud. Appl. Math. 151 (2023) 1230–1281 (J. Chen) School of Mathematics, Southeast University, Nanjing, Jiangsu 210096, PR China Email address: cjb@seu.edu.cn (D. Pelinovsky) Department of Mathematics and...

    2023

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.