Pith. sign in

REVIEW 3 major objections 5 minor 52 references

Local well-posedness for the derivative nonlinear Schr\"odinger equation with nonvanishing boundary conditions

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves unconditional local well-posedness for derivative NLS perturbations around bounded backgrounds at regularity s>3/4.

desk verdict Genuinely new low-regularity result for non-divergence DNLS around non-L2 backgrounds; the main proof is credible but needs referee scrutiny on compressed modified-energy cancellations and the omitted proof of Theorem 1.1. read the letter →

arxiv 2505.20883 v1 pith:ZW6RPF7Y submitted 2025-05-27 math.AP

Luc Molinet , Tomoyuki Tanaka This is my paper
classification math.AP MSC 35Q5535A0135A02
keywords derivativenonlinearSchrödingerequationnonvanishingboundaryconditionsunconditionalwell-posednessenergymethodmodifiedZhidkovspacedarksolitonlowregularity
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 proves that the derivative nonlinear Schrödinger equation is locally well-posed for perturbations of a fixed bounded background function, which may be a non-decaying solution such as a dark soliton. The main result establishes unconditional well-posedness in H^s(R) for s>3/4, including the general non-divergence case where the coefficients satisfy λ≠0,2μ, provided the background satisfies suitable regularity hypotheses and ∂xψ∈L∞(R;L1(R)). This matters because nonvanishing boundary conditions prevent the usual gauge transformation from removing the problematic nonlinear term, and previous low-regularity methods for the non-divergence case stopped at $H^{1}$. As a byproduct, the same method gives local well-posedness in the Zhidkov space for the original equation.

What carries the argument

The central object is a modified energy for the difference w=u1−u2 of two solutions: Eθ(u1,u2,N0)=Σ_{N∈2Z}($N^{{-1+2δ}}$∨$N^{{2θ}}$)|E_N(u1,u2,N0)|, where E_N is the squared $L^{2}$ norm of the N-th Littlewood-Paley piece of w plus three correction terms built from the anti-derivative ∂$x^{{-1}}$, homogeneous frequency projections, and products such as ∂$x^{{-1}}$$P_N^{2}$P_{N1}w̄ ∂xP_{N3}u2 ∂$x^{{-1}}$Ṗ_M(P_{N2}w̄P_{N4}u1). These correction terms cancel exactly the most problematic nonresonant interactions that would otherwise lose a derivative; the leftover terms are either integrated by parts so that the anti-derivative swaps a high derivative for a low one, or controlled through the resonance lower bound |Ω|≳N1M together with refined Strichartz estimates.

What would settle it

One concrete way to falsify the central claim would be to find a background ψ satisfying (1.11) and (6.1) and two distributional solutions u1,u2∈L∞([0,T];H^s(R)) with 3/4<s<1 that have the same initial value but differ at some positive time; Theorem 1.2's unconditional uniqueness asserts that no such pair can exist.

Watch

Extended reading notes

Core claim

The central claim is that the derivative loss and the low×high×high→low resonant interactions caused by terms such as |u|^2∂xu and |ψ|^2∂xu can be controlled at regularity s>3/4 by an energy method with correction terms. For the perturbation equation (1.4), the authors prove existence, uniqueness in the full class L∞([0,T];H^s(R)) without auxiliary spaces, and continuity of the flow map, for every background ψ satisfying (1.11) and, in the non-divergence case, the additional condition ∂xψ∈L∞(R;L1(R)). The proof does not use the gauge transformation and instead treats the equation directly, so it covers general coefficients (λ,μ) and backgrounds such as dark solitons.

Load-bearing premise

The proof depends on the background ψ being smooth enough and nearly a solution: its residual Ψ(ψ) must lie in L∞(R;$H^{{s+ε}}$(R)) and its weighted derivatives must stay bounded, because the energy estimates hand control of every ψ-dependent term to exactly those norms.

Editorial extensions

If this is right

  • If Theorem 1.2 is correct, uniqueness holds in the full class L∞([0,T];H^s(R)) without imposing any auxiliary Bourgain or Strichartz regularity on the solution.
  • The dark soliton profile (1.2) satisfies the stated hypotheses, so the result covers perturbations of an explicit non-decaying solution, and constant backgrounds are also included.
  • The Zhidkov-space statement (Theorem 1.3) extends previous well-posedness results to general coefficients (λ,μ), with s>3/4 in the divergence case and s≥1 otherwise.
  • The threshold s>3/4 is where the products appearing in the nonlinearity become well-defined at the required regularity, so the regularity level is natural rather than accidental.

Reading between the lines

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

  • The condition ∂xψ∈L∞(R;L1(R)) is used specifically to control low-frequency outputs of ∂x^{-1}Ṗ_M(wψ); it may be stronger than necessary, and weighted low-frequency spaces could allow weaker integrability assumptions on ∂xψ.
  • Because the correction-term cancellation is formulated in general terms, the same modified-energy construction could transfer to other derivative-type dispersive equations with nondecaying backgrounds, wherever a similar resonance lower bound holds.
  • The difference estimate in the non-divergence case is performed at H^{s-1/2} rather than H^{s-1}, suggesting that the natural well-posedness space for the non-divergence case is half a derivative lower than for the divergence case.
  • A testable extension would be to push the method toward s=3/4 exactly or to relax the background condition from ∂xψ∈L1 to a Besov-type integrability, in order to see whether the regularity threshold is sharp.
Share X Bluesky LinkedIn Reddit HN

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 paper studies the derivative nonlinear Schrödinger equation (1.1) on the real line with nonvanishing boundary conditions, written as a perturbation equation (1.4) around a background function ψ. The main new result, Theorem 1.2, asserts unconditional local well-posedness in H^s(R) for 3/4<s<1 under the hypotheses (1.11) and ∂xψ∈L∞(R;L1(R)), i.e. for the general non-divergence-form case. The proof is based on the energy method with correction terms (modified energy) developed in Section 6, combined with refined Strichartz estimates from Section 3 and Bourgain-type estimates for nonresonant interactions from Section 4. The paper also states Theorem 1.1 (well-posedness for s>3/4 in the divergence-form case λ=2μ or λ=0, and for s≥1 otherwise) and Theorem 1.3 (well-posedness in Zhidkov spaces).

Significance. If correct, Theorem 1.2 is the first low-regularity unconditional well-posedness result for the general derivative nonlinear Schrödinger equation with nonvanishing boundary conditions, going below H^1 in the non-divergence case. The proof is technically ambitious: it combines refined Strichartz estimates, frequency-envelope continuity arguments, and a modified energy designed to cancel the low×high×high→low interactions that obstruct the H^{s-1} difference estimate. The paper also gives explicit estimates in Propositions 4.11, 5.2, and 6.4, and includes a detailed frequency-envelope proof of continuous dependence in Theorem 1.2. These are real strengths. However, the most delicate part, the modified-energy cancellation in Proposition 6.4, is only partially written out, and one of the main theorems (Theorem 1.1) is stated without proof.

major comments (3)
  1. [§7.1] The manuscript explicitly states 'We omit the proof of Theorem 1.1' and only proves Theorem 1.2. Since Theorem 1.1 is a main result announced in the abstract and the introduction, omitting its proof leaves a load-bearing claim unsupported. The paper should either provide a complete proof of Theorem 1.1, or restate it as a corollary with an explicit argument assembling the estimates of Sections 4 and 5, including the claimed time of existence T ≥ g(∥u0∥_{H^{3/4+}}).
  2. [§6, Proposition 6.4] The key cancellation of the low×high×high→low interactions is only sketched. The reduction of A1+A2+A3+A4 in (6.16) is asserted 'by integration by parts' with the help of (6.15), but the intermediate algebraic steps are not displayed; the residual term involving ξ4ξ1^{-1} is crucial because it is the mechanism that avoids a derivative loss. Similarly, the claimed collapse of A5,3,5 + A5,3,6 in (6.21) into two controllable terms is asserted without showing the integration by parts. Because a sign or boundary-term error in these equalities would leave a derivative loss that is not recovered anywhere else, the proof of the central a priori estimate (6.10) is not checkable from the preprint as written.
  3. [§6, estimates for A6 and A7] Several entire families of estimates in the proof of Proposition 6.4 are deferred with phrases such as 'Estimates for A6 can be obtained similarly to A5' and 'this case follows from the estimates for J(2)_t'. The terms A5, A6, and A7 are not peripheral; they control the nonlinear contributions to the time derivative of the modified energy. Since the unconditional uniqueness step (7.1) depends on the full strength of (6.10), the authors should either provide the complete estimates or explicitly identify which estimates from [37, 38, 40] are being invoked and why they cover these terms without modification.
minor comments (5)
  1. [Title/Abstract] There is a typo in the title of the arXiv version ('DERIV ATIVE' instead of 'DERIVATIVE'); the abstract is otherwise clear.
  2. [§1.5, (1.11) vs §4.2] The hypothesis (1.11) uses J_x^{s+1+ε}ψ, while Lemma 4.6 and Proposition 4.11 sometimes state bounds with J_x^{s+1}ψ; this is harmless since J^{s+1}ψ is controlled by J^{s+1+ε}ψ, but the notation should be harmonized.
  3. [§6, Definition 5 and (6.7)] In the proof of Proposition 6.3, the estimate for E_N^3 uses the bound ∥J_xψ∥_{L∞}, but the statement of the proposition only assumes J_xψ∈L∞; this is consistent, but the display 'N^{2θ}|E_N^3|' omits the explicit factors N_4^{1/2} and the sum over N_4, which would help the reader verify the claimed N^{-s+θ+1/2} decay.
  4. [§7.1, Step 2] In (7.4), the expression '∥u0∥_{L∞_T H^s_x}' should be '∥u0∥_{H^s_x}'; there are also some naming inconsistencies between T3(s,n) and its later use in the definition of eT3(s).
  5. [§7.2] The proof of Theorem 1.3 uses Proposition 5.3, but the proposition is stated for s∈(3/4,1] with λ=2μ and for s=1 otherwise; the proof in Step 3 should explicitly state which range of s is being used and why the assumptions of Theorem 1.3 are satisfied.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found; the derivation is a standard energy-method argument built on stated assumptions and published prior estimates, though it leans heavily on the authors' own earlier work.

full rationale

The central claim, Theorem 1.2, is proved by deriving the modified-energy a priori estimate (6.10) from the equation (1.4) and the stated assumptions (1.11), (6.1). There is no fitted parameter that is later renamed as a prediction: the coefficients c1, c2, c3 in the modified energy are free constants chosen in the proof to cancel explicit interaction terms, which is a legitimate algebraic construction rather than circularity. The assumptions on psi are used to control psi-dependent terms and are satisfied by the exhibited dark-soliton example, so they are not hidden forms of the conclusion. The heavy citations to the authors' prior work, especially [37], [38], [39], and [40], are used as methodological background or as source of published lemmas with independent proofs, not as an unverified premise that assumes the target well-posedness result. The unconditional uniqueness step (7.1) is a Gronwall-type argument from coercivity of the modified energy and the a priori estimate; since the initial data coincide, the right-hand side vanishes. Although the proof of Proposition 6.4 is dense and some cancellations are only sketched with phrases such as 'by integration by parts, we obtain', such gaps are correctness risks rather than self-referential reductions. The paper is therefore not circular in any of the enumerated senses; the score 2 reflects only the unusually heavy reliance on the authors' own prior estimates, none of which is load-bearing in a circular way.

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

The proof is a mathematical derivation with no data fitting. The constants ε, δ, θ and c1,c2,c3 are proof parameters. The main domain assumptions are the conditions on ψ in (1.11) and (6.1). No new physical entities are introduced.

free parameters (4)
  • ε = sufficiently small
    Small constant fixed in (1.11) and throughout to absorb derivative losses; no data fitting.
  • δ = sufficiently small
    Exponent in the low-frequency weight of H^{s,δ}, Definition 5; chosen positive small.
  • θ = any value in (1/4, s/2−1/8)
    Regularity level of the difference in the modified energy, Proposition 6.4; chosen in a nonempty interval.
  • c1, c2, c3 = real constants fixed in proof
    Constants in the modified energy (6.5), chosen to cancel specific resonant interactions; not fitted to data.
assumptions (4)
  • domain assumption ψ satisfies (1.11): J_x^{s+1+ε}ψ∈L∞(R^2), ∂tψ∈L∞(R^2), Ψ∈L∞(R;H^{s+ε}(R))
    Assumed in Theorems 1.1 and 1.2; used throughout the energy estimates to shift derivatives onto ψ and control the residual Ψ.
  • domain assumption For Theorem 1.2, additional ∂xψ∈L∞(R;L1(R)) (6.1)
    Needed in Section 6 to control low-frequency anti-derivative terms via (6.33); not required for Theorem 1.1.
  • standard math Standard Littlewood-Paley theory, Sobolev and Bourgain spaces, Strichartz estimates, frequency envelope method
    Used throughout Sections 2-4 as background tools; not proved in this paper.
  • standard math Decomposition of Y^s functions into C_b^∞ + H^s (Lemma 7.1, quoted from [12])
    Basis for Theorem 1.3; cited from prior literature.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Local well-posedness for the derivative nonlinear Schr\"odinger equation with nonvanishing boundary conditions." pith.science (2026). https://pith.science/paper/ZW6RPF7Y

@misc{pith2026250520883,
  author       = {Pith},
  title        = {Pith review of: Local well-posedness for the derivative nonlinear Schr\"odinger equation with nonvanishing boundary conditions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZW6RPF7Y}},
  note         = {Machine review of arXiv:2505.20883}
}
abstract

We consider the derivative nonlinear Schr\"odinger equation on the real line, with a background function $\psi(t,x)\in L^\infty(\mathbb{R}^2)$ that satisfies suitable conditions. Such a function may, for example, be a non-decaying solution of the equation, such as a dark soliton. By developing the energy method with correction terms, we prove that the Cauchy problem for perturbations around such an $L^\infty$ function is unconditionally locally well-posed in $ H^s(\mathbb{R}) $ for $ s>3/4 $. As a byproduct, we also establish local well-posedness in the Zhidkov space.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

52 extracted references · 51 canonical work pages

  1. [1]

    Abdallah, On the well-posedness of dispersive–dissipative one dimensional equations with non decaying initial data , Monatsh

    M. Abdallah, On the well-posedness of dispersive–dissipative one dimensional equations with non decaying initial data , Monatsh. Math. 203 (2024), 267–311

    work page 2024

  2. [2]

    G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, New York, 2019

    work page 2019

  3. [3]

    Babin, A

    A. Babin, A. Ilyin, and E. Titi, On the regularization mechanism for the periodic Korteweg-de Vries equation, Comm. Pure Appl. Math. 64 (2011), 591–648

    work page 2011

  4. [4]

    Bahouri and G

    H. Bahouri and G. Perelman, Global well-posedness for the derivative nonlinear Schr¨ odinger equation, Invent. Math. 229 (2022), 639–688

    work page 2022

  5. [5]

    H. A. Biagioni and F. Linares, Ill-posedness for the derivative Schr¨ odinger and generalized Benjamin-Ono equations, Trans. Amer. Math. Soc. 353 (2001), 3649–3659

    work page 2001

  6. [6]

    J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and appli- cations to nonlinear evolution equations, part I: Schr¨ odinger equation, part II: The KdV- equation, Geom. Funct. Anal. 3 (1993), 107–156, 209–262

    work page 1993

  7. [7]

    Cazenave, Semilinear Schr¨ odinger Equations, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, 2003

    T. Cazenave, Semilinear Schr¨ odinger Equations, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, 2003

    work page 2003

  8. [8]

    Colliander, M

    J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global well-posedness for Schr¨ odinger equations with derivative, SIAM J. Math. Anal. 33 (2001), 649–669

    work page 2001

Show all 52 references
  1. [9]

    Colliander, M

    J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, A refined global well-posedness result for Schr¨ odinger equations with derivative, SIAM J. Math. Anal. 34 (2002), 64–86

    2002

  2. [10]

    Ehrstr¨ om and Y

    M. Ehrstr¨ om and Y. Wang,Enhanced existence time of solutions to the fractional Korteweg–de Vries equation, SIAM J. Math. Anal. 51 (2019), 3298–3323

    2019

  3. [11]

    Fukaya, M

    N. Fukaya, M. Hayashi, and T. Inui, A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schr¨ odinger equation, Anal. PDE 10 (2017), 1149–1167

    2017

  4. [12]

    Gallo, Korteweg-de Vries and Benjamin-Ono equations on Zhidkov spaces, Adv

    C. Gallo, Korteweg-de Vries and Benjamin-Ono equations on Zhidkov spaces, Adv. Differential Equations 10 (2005), 277–308

    2005

  5. [13]

    Ginibre, Y

    J. Ginibre, Y. Tsutsumi, and G. Velo, On the Cauchy problem for the Zakharov system , J. Funct. Anal. 151 (1997), 384–436

    1997

  6. [14]

    Grafakos and S

    L. Grafakos and S. Oh, The Kato-Ponce inequality , Comm. Partial Differential Equations 39 (2014), 1128–1157

    2014

  7. [15]

    Z. Guo, S. Kwon, and T. Oh, Poincar´ e-Dulac normal form reduction for unconditional well- posedness of the periodic cubic NLS , Comm. Math. Phys. 322 (2013), 19–48. 82 L. MOLINET AND T. TANAKA

    2013

  8. [16]

    Harrop-Griffiths, R

    B. Harrop-Griffiths, R. Killip, M. Ntekoume, and M. Vi¸ san, Global well-posedness for the derivative nonlinear Schr¨ odinger equation inL2(R), to appear in J. Eur. Math. Soc. (JEMS)

  9. [17]

    Hayashi, The initial value problem for the derivative nonlinear Schr¨ odinger equation in the energy space, Nonlinear Anal

    N. Hayashi, The initial value problem for the derivative nonlinear Schr¨ odinger equation in the energy space, Nonlinear Anal. 20 (1993), 823–833

    1993

  10. [18]

    Hayashi and T

    N. Hayashi and T. Ozawa, On the derivative nonlinear Schr¨ odinger equation , Phys. D 55 (1992), 14–36

    1992

  11. [19]

    Herr, On the Cauchy problem for the derivative nonlinear Schr¨ odinger equation with periodic boundary condition, Int

    S. Herr, On the Cauchy problem for the derivative nonlinear Schr¨ odinger equation with periodic boundary condition, Int. Math. Res. Not. (2006), Art. ID 96763, 33 pp

    2006

  12. [20]

    A. D. Ionescu, C. E. Kenig, and D. Tataru, Global well-posedness of the initial value problem for the KP-I equation in the energy space , Invent. Math. 173 (2008), 265–304

    2008

  13. [21]

    Iorio, F

    R. Iorio, F. Linares, and M. Scialom, KdV and BO equations with bore-like data , Differential Integral Equations 11 (1998), 895–915

    1998

  14. [22]

    Jenkins, J

    R. Jenkins, J. Liu, P. Perry, and C. Sulem, Soliton resolution for the derivative nonlinear Schr¨ odinger equation, Comm. Math. Phys. 363 (2018), 1003–1049

    2018

  15. [23]

    Kato, On nonlinear Schr¨ odinger equations

    T. Kato, On nonlinear Schr¨ odinger equations. II. H s-solutions and unconditional well- posedness, J. Anal. Math. 67 (1995), 281–306. with Correction in J. Anal. Math 68 (1996), 305

    1995

  16. [24]

    D. J. Kaup and A. C. Newell, An exact solution for a derivative nonlinear Schr¨ odinger equa- tion, J. Math. Phys. 19 (1978), 798–801

    1978

  17. [25]

    Killip and M

    R. Killip and M. Vi¸ san,KdV is well-posed in H −1, Ann. of Math. 190 (2019), 249–305

    2019

  18. [26]

    Kishimoto, Unconditional uniqueness of solutions for nonlinear dispersive equations , arXiv:1911.04349v4

    N. Kishimoto, Unconditional uniqueness of solutions for nonlinear dispersive equations , arXiv:1911.04349v4

    1911 arXiv

  19. [27]

    Kishimoto and Y

    N. Kishimoto and Y. Tsutsumi, Gauge transformation for the kinetic derivative nonlinear Schr¨ odinger equation on the torus, arXiv:2303.17359v2

    arXiv

  20. [28]

    Klainerman and M

    S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), 1221–1268

    1993

  21. [29]

    Koch and N

    H. Koch and N. Tzvetkov, On the local well-posedness of the Benjamin-Ono equation in H s(R), Int. Math. Res. Not. IMRN 26 (2003), 1449–1464

    2003

  22. [30]

    Lannes, The water waves problem, Math

    D. Lannes, The water waves problem, Math. Surveys Monogr., 188, American Mathematical Society, Providence, 2013

    2013

  23. [31]

    Le Coz and T.-P

    S. Le Coz and T.-P. Tsai, Finite and infinite soliton and kink-soliton trains of nonlinear Schr¨ odinger equations, Proceedings of the Sixth International Congress of Chinese Mathe- maticians. Vol. I, 43–56, Adv. Lect. Math. (ALM), 36, Int. Press, Somerville, MA, 2017

    2017

  24. [32]

    Linares and G

    F. Linares and G. Ponce, Introduction to nonlinear dispersive equations, Universitext, Springer, New York, 2015

    2015

  25. [33]

    Masmoudi and K

    N. Masmoudi and K. Nakanishi, Energy convergence for singular limits of Zakharov type systems, Invent. Math. 172 (2008), 535–583

    2008

  26. [34]

    K. Mio, T. Ogino, K. Minami, and S. Takeda, Modified nonlinear Schr¨ odinger equation for Alfv´ en waves propagating along the magnetic field in cold plasmas , J. Phys. Soc. Jpn. 41 (1976), 265–271

    1976

  27. [35]

    Mjølhus, On the modulational instability of hydromagnetic waves parallel to the magnetic field

    E. Mjølhus, On the modulational instability of hydromagnetic waves parallel to the magnetic field. J. Plasma Phys. 16 (1976), 321–334. DNLS WITH NONV ANISHING BOUNDARY CONDITIONS 83

    1976

  28. [36]

    Molinet, J.-C

    L. Molinet, J.-C. Saut, and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations, SIAM J. Math. Anal. 33 (2001), 982–988

    2001

  29. [37]

    Molinet and S

    L. Molinet and S. Vento, Improvement of the energy method for strongly nonresonant dispersive equations and applications , Anal. PDE 8 (2015), 1455–1495

    2015

  30. [38]

    Molinet, D

    L. Molinet, D. Pilod, and S. Vento, On well-posedness for some dispersive perturbations of Burgers’ equation, Ann. I. H. Poincar´ e Anal. Non Lin´ eaire35 (2018), 1719–1756

    2018

  31. [39]

    Molinet, D

    L. Molinet, D. Pilod and S. Vento, On unconditional well-posedness for the periodic modified Korteweg-de Vries equation, J. Math. Soc. Japan 71 (2019), 147–201

    2019

  32. [40]

    Molinet and T

    L. Molinet and T. Tanaka, Unconditional well-posedness for some nonlinear periodic one- dimensional dispersive equations , J. Funct. Anal. 283 (2022), 109490

    2022

  33. [41]

    Mosincat and H

    R. Mosincat and H. Yoon, Unconditional uniqueness for the derivative nonlinear Schr¨ odinger equation on the real line , Discrete Contin. Dyn. Syst. 40 (2020), 47–80

    2020

  34. [42]

    Ozawa, On the nonlinear Schr¨ odinger equations of derivative type, Indiana Univ

    T. Ozawa, On the nonlinear Schr¨ odinger equations of derivative type, Indiana Univ. Math. J. 45 (1996), 137–163

    1996

  35. [43]

    Palacios, Local well-posedness for the gKdV equation on the background of a bounded func- tion, Rev

    J. Palacios, Local well-posedness for the gKdV equation on the background of a bounded func- tion, Rev. Math. Iberoam. 39 (2023), 341–396

    2023

  36. [44]

    Takaoka, Well-posedness for the one-dimensional nonlinear Schr¨ odinger equation with the derivative nonlinearity, Adv

    H. Takaoka, Well-posedness for the one-dimensional nonlinear Schr¨ odinger equation with the derivative nonlinearity, Adv. Differential Equations 4 (1999), 561–580

    1999

  37. [45]

    Tao, Global well-posedness of the Benjamin-Ono equation in H 1(R), J

    T. Tao, Global well-posedness of the Benjamin-Ono equation in H 1(R), J. Hyperbolic Differ. Equ. 1 (2004), 27–49

    2004

  38. [46]

    Tao, Nonlinear dispersive equations, CBMS Reg

    T. Tao, Nonlinear dispersive equations, CBMS Reg. Conf. Ser. Math., 106, American Mathe- matical Society, Providence, 2006

    2006

  39. [47]

    Tsutsumi and I

    M. Tsutsumi and I. Fukuda, On solutions of the derivative nonlinear Schr¨ odinger equation. Existence and uniqueness theorem , Funkcial. Ekvac. 23 (1980), 259–277

    1980

  40. [48]

    Tsutsumi and I

    M. Tsutsumi and I. Fukuda, On solutions of the derivative nonlinear Schr¨ odinger equation, II, Funkcial. Ekvac. 24 (1981), 85–94

    1981

  41. [49]

    Van Tin, On the Cauchy problem for a derivative nonlinear Schr¨ odinger equation with nonvanishing boundary conditions , Evol

    P. Van Tin, On the Cauchy problem for a derivative nonlinear Schr¨ odinger equation with nonvanishing boundary conditions , Evol. Equ. Control Theory 11 (2022), 837–867

    2022

  42. [50]

    Y. Y. S. Win, Unconditional uniqueness of the derivative nonlinear Schr¨ odinger equation in energy space, J. Math. Kyoto Univ. 48 (2008), 683–697

    2008

  43. [51]

    Wu, Global well-posedness on the derivative nonlinear Schr¨ odinger equation, Anal

    Y. Wu, Global well-posedness on the derivative nonlinear Schr¨ odinger equation, Anal. PDE 8 (2015), 1101–1112

    2015

  44. [52]

    P. E. Zhidkov, Korteweg-de Vries and nonlinear Schr¨ odinger equations: qualitative theory, Lecture Notes in Mathematics, 1756, Springer-Verlag, Berlin, 2001. (L. Molinet) Institut Denis Poisson, Universit ´e de Tours, Universit ´e d’Orl ´eans, CNRS, Parc Grandmont, 37200 Tour...

    2001

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.