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Large-time asymptotics for the defocusing Manakov system on a nonzero background

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Pith's one-line read The paper derives the first long-time asymptotic formula in the soliton region for a two-component nonlinear Schrödinger system on a nonzero background: a modulated multisoliton plus an explicit t^{-1/2} dispersive correction.

desk verdict First real long-time asymptotic theorem for the defocusing Manakov system with nonzero background; the proof is serious but long, with a few verification gaps that need closing. read the letter →

arxiv 2512.21841 v2 pith:NFGQD2KM submitted 2025-12-26 nlin.SI math.AP

Xianguo Geng , Haibing Zhang , Jiao Wei This is my paper · ORCID
classification nlin.SImath.AP MSC 35Q5535Q1537K15
keywords Manakovsystemdefocusingnonzeroboundaryconditionslong-timeasymptoticsRiemann-Hilbertproblemsolitonregionradiationtermmodulatedmultisoliton
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 aims to prove that the defocusing Manakov system—a two-component vector nonlinear Schrödinger equation—on a nonzero background has, in the space-time soliton region, a universal long-time shape: a modulated multisoliton plus a radiation term of order t^{-1/2} with an explicit coefficient, up to O(t^{-1} log t). This would be the first long-time asymptotic theorem in the soliton region for any vector NLS system with nonzero background. The result matters because it shows that the vector coupling creates a dispersive correction that has no analog in the scalar defocusing NLS equation; when the extra reflection coefficients vanish, the correction disappears and the formula reduces to the known scalar result. The proof works by setting up a 3x3 matrix Riemann-Hilbert problem and carrying out a nonlinear steepest descent analysis on it.

What carries the argument

The central machinery is a 3x3 matrix Riemann-Hilbert problem whose jump matrix is factored, conjugated, and deformed through a chain of transformations until it becomes a small-norm problem. The outer model is a modulated pure-soliton Riemann-Hilbert problem, while local models near the two saddle points z0 = q0^2/xi and z1 = xi are solved explicitly in terms of parabolic cylinder functions. The diagonal factors delta1, delta, and delta^sharp, built from the reflection coefficients through trace formulas, encode the scattering data and are responsible for the explicit t^{-1/2} radiation coefficient.

What would settle it

Take a one-soliton initial datum with nonzero vector reflection coefficients, simulate or compute the solution at large times in a fixed velocity window, and compare the residual q(x,t) - q_msol^{[1]}(x,t) to q_rad(x,t)/sqrt(t) with q_rad given by formulas (1.7)-(1.8); if the t^{-1/2} term is absent or has a different coefficient, the theorem is false.

Watch

Extended reading notes

Core claim

For the defocusing Manakov system with parallel nonzero boundary conditions and initial data that equal the backgrounds outside a compact set, the solution in the soliton region |x/(2t)| < q0 is shown to satisfy q(x,t) = q_msol^[N](x,t) + q_rad(x,t)/sqrt(t) + O(t^{-1} log t), uniformly on compact velocity intervals. The leading term q_msol^[N] is a modulated N-soliton solution built from the discrete scattering data, and q_rad is an explicit, x-dependent radiation coefficient expressed in terms of scattering data and solutions of model Riemann-Hilbert problems; separate formulas are given for the right and left halves of the soliton region. The t^{-1/2} term is genuinely a vector effect: whe

Load-bearing premise

The proof requires the initial data to equal the two constant backgrounds exactly outside a finite interval, so that the reflection coefficients are analytic in a neighborhood of the real axis; without this analyticity, the contour deformations and small-norm error estimates do not apply as written.

Editorial extensions

If this is right

  • In the soliton region, the long-time solution of the defocusing Manakov system is a modulated multisoliton with an explicitly computable t^{-1/2} radiation correction.
  • The t^{-1/2} correction is a vector coupling effect: setting the vector reflection coefficients to zero makes it vanish and recovers the scalar defocusing NLS asymptotics.
  • The asymptotic formulas for the right and left halves of the soliton region agree near xi = 0, so there is no transition zone between them.
  • Only solitons whose velocities lie within the observation window contribute to the leading term; the others contribute terms exponentially small in t.
  • The error term O(t^{-1} log t) is uniform on compact velocity intervals inside the soliton region.
  • The method is designed so that the final Riemann-Hilbert problem has no singular behavior at the origin or branch points, which is necessary for the small-norm analysis to close.

Reading between the lines

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

  • Editorial extension: the same 3x3 block structure is likely extendable to the N-component defocusing NLS system, where the authors suggest a parallel derivation; this could be tested by working out the N=3 case explicitly.
  • Editorial extension: the explicit t^{-1/2} coefficient may provide a quantitative signature of vector coupling in experiments or numerical simulations, since it is absent in the scalar equation and depends on both components of the background.
  • Editorial extension: the compact-support assumption is probably removable using a bar-dbar steepest descent method; if so, the same leading asymptotics should hold for Sobolev-class initial data with the same explicit radiation term.
  • Editorial extension: the transition region |x/(2t)| near q0, where the authors anticipate Painlevé II asymptotics, may also inherit a vector-specific correction that could be checked against direct simulation.
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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

4 major / 4 minor

Summary. The paper develops the Deift–Zhou steepest descent method for the defocusing Manakov system with parallel nonzero boundary conditions. Under Assumptions 1.1 (smooth initial data compactly supported away from the background, finitely many simple discrete eigenvalues on |z|=q0 with τj/ζj<0, and generic branch-point behavior), it proves Theorem 1.6: in the right and left soliton regions |x/(2t)|<q0, the solution equals a modulated N-soliton plus an explicit O(t^{-1/2}) radiation term, with uniform error O(t^{-1} ln t). Theorem 1.7 removes solitons whose velocities lie outside the compact velocity interval. The proof imports the 3x3 Riemann–Hilbert formulation from Biondini–Kraus, applies a chain of transformations, constructs outer and local (parabolic-cylinder) model problems, and reduces the final error to a small-norm RH problem. The radiation coefficient is traced to the two stationary points and is claimed to vanish in the scalar limit r1=r3=0, recovering the defocusing NLS result of Cuccagna–Jenkins.

Significance. If correct, this is the first long-time asymptotic theorem for a vector NLS system with nonzero background in the soliton region. The explicit t^{-1/2} dispersive correction and its vanishing in the scalar limit is a genuinely new structural result. The paper presents a very detailed transformation chain, explicit local model formulations in Appendix C, and a self-contained solvability proof for the modulated soliton RH problem in Appendix B. The explicit formulas (1.8), (1.11), the uniformity statements, and the reduction to the scalar case in Remark 1.10 are notable strengths. The main caveat is that several algebraic verifications that are load-bearing for the proof are asserted rather than fully shown, in particular the full simplification of the jump matrix and the branch-point cancellations. These gaps appear fillable, but they are central enough that the manuscript needs revision before the result can be regarded as fully verified.

major comments (4)
  1. [Appendix A.2, Eq. (2.19)] The simplification of the jump matrix is the foundation of the entire Deift–Zhou analysis. The derivation verifies only the (1,1) and (1,2) entries and then states that the remaining entries follow by a similar computation. All nine entries of (2.19) enter the factorizations (3.2), (3.17), (3.19), and the subsequent contour deformations. The symmetries (2.8)–(2.10) are nontrivial, and the entries mix r1, r2, r3 with 1/γ and conjugation. A sign or coefficient error in an unshown entry would change the triangular factorizations and invalidate Theorem 1.6. Please provide a complete entry-by-entry derivation or a rigorous symmetry argument that determines all entries from the two shown.
  2. [§3.7, Lemma 3.16; §3.5, Lemma 3.8] The boundedness of the final error E at z=0 and z=±q0 is load-bearing for the small-norm RH problem. In Lemma 3.16, the proof for z=0 is only a claim about the structure (3.73) after 'examining all transformations,' and the proof at q0 relies on an unstated direct calculation combining (3.75) and (3.76); the same statement for -q0 is dismissed as analogous. Lemma 3.8 likewise contains a 'straightforward calculation' to verify the symmetry of F at q0, after some limits involving a12,+/a11,+. Any missed pole at these points would destroy the L^p estimates in Lemma 3.20. These cancellations must be shown in sufficient detail, either in the main text or in an appendix.
  3. [§3.5, Lemma 3.6; §4.2, Lemma 4.1] The endpoint estimates (3.37)–(3.38) and (4.9), stated to follow from 'relatively straightforward estimates,' are used in Lemmas 3.14–3.15 and 4.4 to obtain the local-parametrix error O(t^{-1/2} ln t). These estimates control the singular behavior of δ(z), δ~(z), and their left-sector counterparts near the stationary points. Since the stated error term of Theorem 1.6 depends on these logarithmic bounds, the proof should either be included or cited to a specific lemma in the literature with the hypotheses verified.
  4. [§3.5, Lemmas 3.7–3.8; §3.7, Lemma 3.17] Several residue and analyticity properties after the transformations are asserted without calculation or deferred by analogy. Lemma 3.7 gives residue conditions after the sixth transformation but the proof is omitted; Lemma 3.17 defers the analyticity of E at the discrete spectrum to an analogy with Appendix A.3. These properties are needed to ensure that the final small-norm problem is well-defined and that E has no poles in the regions where it is supposed to be analytic. Please expand these arguments or move them to an appendix with all steps.
minor comments (4)
  1. [Abstract and §1] The abstract and introduction state the result without emphasizing the compact-support assumption in Assumptions 1.1. The theorem is proved only for this class, and Remark 1.2 explicitly says the extension is only conjectured. Please qualify the statements in the abstract and introduction accordingly.
  2. [Remark 1.9] The compatibility of the left and right asymptotic formulas near ξ=0 is asserted with 'it can be proved,' but no proof is given. Since this statement is not used in the proof of the main theorem, it should either be proved in an appendix or explicitly labeled as a conjecture.
  3. [Appendix B] The proof of unique solvability of RH problem 3.11 solves only the M23 equation and then states that M13 and M33 are obtained by 'completely analogous' computations. The right-hand sides for the different columns are not identical, so a brief indication of how the same positive-definite argument applies would improve rigor.
  4. [Throughout] There are several typos and notational slips: 'vecter' in §1; 'M mosl' in (3.52); the inconsistent use of R_+/- and ℛ_sol, and the figure captions are not fully explicit about orientation. These should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main asymptotics are derived from an external Riemann–Hilbert formulation and standard Deift–Zhou analysis, with no fitted parameters and no load-bearing self-citation chain.

full rationale

The paper's central result, Theorem 1.6, is a genuinely derived asymptotic expansion rather than a repackaged input. The Riemann–Hilbert problem (RH Problem 2.3) is imported from Biondini–Kraus [17], an external source, and the subsequent Deift–Zhou steepest-descent analysis is explicit: each transformation M -> M^(1) -> ... -> E is computed from the scattering data, and the t^{-1/2} coefficient q_rad is obtained from the model Riemann–Hilbert problems at the stationary points z0 and z1 via parabolic-cylinder asymptotics (Appendix C), not from fitting. No parameter appearing in the final asymptotic formula is fitted to the quantity being predicted; the functions delta_1, delta, delta^sharp, L_A, L_B are all defined from the initial data's scattering data through Plemelj formulas and trace identities. The scalar reduction check in Remark 1.10 (r1 = r3 = 0 eliminates the t^{-1/2} term) is a consistency check, not a circular input. The only self-references are contextual ([5], [22], [49]) or a forward-looking companion paper [50] on the transition region, which is explicitly described as separate work and is never used to justify Theorem 1.6. The partially verified simplification of the jump matrix in Appendix A.2, where only the (1,1) and (1,2) entries are computed in detail and the rest are asserted to follow similarly, is a real completeness/correctness risk, but it is not a circularity: (2.19) is derived from the external [17] jump matrix, and the derivation does not presuppose the asymptotic result. Thus the paper is self-contained for its main claim, and no circular step was identified.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No new physical entities, forces, or dimensions are introduced. The free parameters listed are harmless analytic partitions, not fitted quantities. The mathematical burden is carried by the imported IST formulation from [17] and by four stated domain assumptions on the initial data and scattering data.

free parameters (2)
  • soliton-region bounds m0, m1 = arbitrary constants, 0 < m0 < m1 < q0
    Chosen by hand to define compact velocity intervals I±; the theorem is uniform in ξ over any such interval, so these do not affect the scientific content.
  • strip width ε = small positive constant
    Selected small enough to avoid discrete spectrum in the contour deformation; final estimates are independent of the precise value.
assumptions (5)
  • domain assumption The inverse scattering/Riemann-Hilbert characterization of the defocusing Manakov system with NZBC from [17] is correct and applicable.
    The entire paper builds on the RH problem 2.3, Jost solutions, scattering matrix symmetries, and reconstruction formula taken from Biondini-Kraus [17].
  • domain assumption Initial perturbation q(x,0)-q± is identically zero outside a compact set.
    Assumptions 1.1 and Remark 1.2; this makes reflection coefficients analytic off the real axis and removes the need for bar-d-bar analysis.
  • domain assumption The scattering coefficient a11 has only finitely many simple zeros on |z|=q0 and τ_j/ζ_j < 0.
    Assumptions 1.1 and Eq. (1.3); needed for regular, globally defined soliton solutions and for unique solvability of the modulated pure-soliton RH problem in Appendix B.
  • domain assumption Branch-point behavior lim_{z→±q0} (z∓q0)a11(z) ≠ 0, Eq. (1.4).
    Used to control the singularities of the outer parametrix and to prove E is well-defined near ±q0 in Lemma 3.16.
  • standard math Standard Plemelj, Cauchy-operator, small-norm RH, and parabolic cylinder asymptotics are valid where invoked.
    These are standard tools used in Sections 3-5 and Appendices A-C; no modification is introduced.

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Pith. "Pith review of Large-time asymptotics for the defocusing Manakov system on a nonzero background." pith.science (2026). https://pith.science/paper/NFGQD2KM

@misc{pith2026251221841,
  author       = {Pith},
  title        = {Pith review of: Large-time asymptotics for the defocusing Manakov system on a nonzero background},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NFGQD2KM}},
  note         = {Machine review of arXiv:2512.21841}
}
abstract

The Manakov system is a two-component nonlinear Schr\"odinger equation. In this paper, we derive a long-time asymptotic formula for the solution of the defocusing Manakov system with nonzero boundary conditions and provide a detailed proof. We first formulate the inverse problem as a $3\times3$ matrix Riemann--Hilbert problem. We then carry out the Deift--Zhou steepest descent analysis for this Riemann--Hilbert problem and obtain the long-time asymptotics in the space-time soliton region. In this region, the leading order of the solution takes the form of a modulated multisoliton. Apart from the error term, we also discover that the defocusing Manakov system has a dispersive correction term of order $t^{-1/2}$, but this term does not exist in the scalar case, and we provide the explicit expression for this dispersion term.

Figures

Figures reproduced from arXiv: 2512.21841 by the authors.

Figure 1
Figure 1. From left to right: The signature tables for [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. The contour Σ(1) and the regions {Rj} 2 j=1. 3.1 The transformation: M → M(1) Lemma 2.2 shows that as z → 0 within the region Sd, the limits of {rj (z)} 3 j=1 exist. This indicates that our augmented contour near zero should be chosen within Sd. More precisely, we define the contour Σ(1) as Σ (1) 2 = n z : (Re z) 2 + (Im z − 1/2)2 = 1/4, − p |ε 2 − ε| ≤ Re z ≤ 0, 0 ≤ Im z ≤ ε o ∪ n z : s + iε, s ≤ −p |ε 2 − ε| o , Σ… view at source ↗
Figure 3
Figure 3. The contour Σ(3) and regions {Ωj} 2 j=1. Lemma 3.5. For each 0 ≤ j ≤ N − 1, M(2)(x, t, z) satisfies the following residue condition at ζj : Resz=ζjM(2)(x, t, z) = lim z→ζj M(2)(x, t, z)   0 0 0 0 0 0 τ˜je θ31(x,t,ζj ) 0 0   , (3.26) where τ˜j = τj δ1(ζj ) δ1(ζ ∗ j ) = τj |δ1(ζj )| 2 . The proof of this lemma is a straightforward calculation and is omitted for brevity. 3.3 The transformations: M(2) → M(3) → M(4) … view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: The contour Σ(4) and regions Ω±. on (z1, +∞). Let’s define the fourth transformation as follows: M(4)(x, t, z) = M(3)(x, t, z)F(z), F(z) =    e Θ   1 − 1 γ(z) r˜ ∗ 1 (z ∗ ) 0 0 1 0 0 0 1   e −Θ, z ∈ Ω+, e Θ   1 0 0 −r˜1(z) 1 0 0 0 1  …
Figure 5
Figure 5. Figure 5: The contour Σ(5) and the regions {Dj} 6 j=1. triangular factorizations of V (5) 1 and V (5) 10 are no longer suitable. Because such factorizations cannot ensure the extended jump matrix asymptotically approaches the identity matrix. This implies that a new transformati…
Figure 6
Figure 6. Figure 6: The countor Σ(7) and the regions {Uj} 5 j=1. Finally, combining the above with (3.46), a straightforward calculation yields Π(q0)F˜ +(q0)Π(q0) = F˜ −(q0). Therefore, we have completed the proof of the first equality in (3.45). Now, V (6) 1 and V (6) 10 are already in t…
Figure 7
Figure 7. Figure 7: From left to right: The signature tables for [PITH_FULL_IMAGE:figures/full_fig_p039_7.png]
Figure 8
Figure 8. Figure 8: The contour Γ(1) and regions {Ω˜ j} 2 j=1. Throughout this section, we let ξ = x/(2t) ∈ I−, and still denote by z1 = ξ the stationary point of the phase function ϕ21, and by z0 = q 2 0 ξ the stationary point of the phase function ϕ32. The signature tables for Re ϕ21(ξ,…
Figure 9
Figure 9. Figure 9: The contour Γ(2) and the regions {D˜ j} 6 j=1. V (1) 2 =   1 0 0 0 1 0 0 − 1 γ(z) r ∗ 3 (z ∗ ) 1   e −Θ, V (1) 4 = eΘ   1 0 0 0 1 −r3(z) 0 0 1   e −Θ, V (1) 3 = eΘ   1 − 1 γ r ∗ 1 −r ∗ 2 0 1 0 0 0 1     1 0 0 r1 1 0 r2 0 1   e −Θ. The remaining jumps re…
Figure 10
Figure 10. Figure 10: The contour Γ(3) . where ˆz = q 2 0 z and δ ♯ (z) = exp 1 2πi Z z0 −∞ ln(1 + 1 γ(s) |r3(s)| 2 ) s − z ds  . (4.4) A direct computation yields 0 < c < 1 + 1 γ(z) |r3(z)| 2 < 1, for all z ∈ Γ (3) 1 and all ξ ∈ I−. Then by the Plemelj formula, δ ♯ satisfies the followi…
Figure 11
Figure 11. Figure 11: The contour Γ(5) and regions {U˜ j} 7 j=1. V (4) 12 = (∆♯P) −1 (z)eΘ   1 0 0 −r˜1 1 0 r˜2 1 γ r˜ ∗ 3 1     1 1 γ r˜ ∗ 1 −r˜ ∗ 2 0 1 ˜r3 0 0 1   e −Θ∆♯ (z)P(z), V (4) 9 = (T ♯ ) −1V (3) 9 T ♯ , V (4) 11 = (T ♯ ) −1V (3) 11 T ♯ . In the last expression above, {r…
Figure 12
Figure 12. Figure 12: The jump contour X = ∪ 4 j=1Xj and the sectors {Oj} 6 j=1. Riemann-Hilbert Problem C.1. The 3 × 3 matrix-valued function MX,L(ζ, y1) satisfies the following properties: 1. MX,L(· , y1) : C \ X → C 3×3 is analytic for ζ ∈ C \ X. 2. The function MX,L(ζ, y1) is continuou…

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Works this paper leans on

62 extracted references · 3 linked inside Pith

  1. [50]

    In preparation (2025)

    Geng, X.G., Zhang, H.B., Painlev´ e-type asymptotics for the defocusing Manakov system under nonzero background. In preparation (2025). 66

    2025

  2. [1]

    SIAM, Philadelphia (1981)

    Ablowitz, M.J., Segur, H., Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)

    1981

  3. [2]

    Cambridge University Press, Cambridge (2003)

    Infeld, E., Rowlands, G., Nonlinear Waves, Solitons and Chaos. Cambridge University Press, Cambridge (2003)

    2003

  4. [3]

    Springer, NewYork (1999)

    Sulem, C., Sulem, P.L., The NonlinearSchr¨ odinger Equation, Self-Focusing and Wave Collapse. Springer, NewYork (1999)

    1999

  5. [4]

    Whitham, G.B., Linear and Nonlinear Waves, Wiley-Interscience, NewYork (1999)

    1999

  6. [5]

    Geng, X.G., Wang, K.D., Chen, M.M., Long-time asymptotics for the spin-1 Gross-Pitaevskii equation. Commum. Math. Phys.382, 585–611 (2021)

    2021

  7. [6]

    Zakharov, V.E., Ostrovsky, L.A., Modulation instability: the beginning. Phys. D238, 540–548 (2009)

    2009

  8. [7]

    Biondini, G., Fagerstrom, E., The integrable nature of modulational instability. SIAM J. Appl. Math.75, 136–163 (2015)

    2015

Show all 62 references
  1. [8]

    SIAM Rev.60, 888–908 (2018)

    Biondini, G., Li, S., Mantzavinos, D., Trillo, S., Universal behavior of modulationally unstable media. SIAM Rev.60, 888–908 (2018)

    2018

  2. [9]

    Kibler, B., Fatome, J., Finot, C., et al., The Peregrine soliton in nonlinear fibre optics. Nat. Phys.6, 790–795 (2010)

    2010

  3. [10]

    Soviet Phys

    Zakharov, V.E., Shabat, A.B., Interaction between solitons in a stable medium. Soviet Phys. JETP37, 823 (1973)

    1973

  4. [11]

    Faddeev, L.D., Takhtajan L.A., Hamiltonian Methods in the Theory of Solitons, Springer Berlin (1987)

    1987

  5. [12]

    Demontis, F., Prinari, B., vanderMee, C., Vitale, F., The inverse scattering transform for the defocusing nonlinear Schr¨ odinger equations with nonzero boundary conditions. Stud. Appl. Math.131, 1–40 (2013)

    2013

  6. [13]

    Biondini, G., Kovaˇciˇc, G., Inverse scattering transform for the focusing nonlinear Schr¨ odinger equation.with nonzero boundary conditions. J. Math. Phys.55, 031506 (2014)

    2014

  7. [14]

    Soviet Phys

    Manakov, S.V., On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Soviet Phys. JETP38, 248–253 (1974)

    1974

  8. [15]

    Cambridge University Press, Cambridge (2004)

    Ablowitz, M.J., Prinari, B., Trubatch, A.D., Discrete and Continuous Nonlinear Schr¨ odinger Systems. Cambridge University Press, Cambridge (2004)

    2004

  9. [16]

    Prinari, B., Ablowitz, M.J., Biondini, G., Inverse scattering transform for the vector nonlin- ear Schr¨ odinger equation with nonvanishing boundary conditions. J. Math. Phys.47, 063508 (2006)

    2006

  10. [17]

    Biondini, G., Kraus, D., Inverse scattering transform for the defocusing Manakov system with nonzero boundary conditions. SIAM J. Math. Anal.47, 706–757 (2015)

    2015

  11. [18]

    Stud.Appl.Math.55, 9-44 (1976)

    Kaup, D.J., The three-wave interaction a nondispersive phenomenon. Stud.Appl.Math.55, 9-44 (1976). 64

    1976

  12. [19]

    Nonlinearity28, 3101–3151 (2015)

    Kraus, D., Biondini, G., Kovaˇciˇc, The focusing Manakov system with nonzero boundary con- ditions. Nonlinearity28, 3101–3151 (2015)

    2015

  13. [20]

    Prinari, B., Biondini, G., Trubatch, A.D., Inverse scattering transform for the multi-component nonlinear Schr¨ odinger equation with nonzero boundary conditions. Stud. Appl. Math.126, 245–302 (2011)

    2011

  14. [21]

    Biondini, G., Kraus, D., Prinari, B., The three-component defocusing nonlinear Schr¨ odinger equation with nonzero boundary conditions. Commun. Math. Phys.348, 475–533 (2016)

    2016

  15. [22]

    Liu, H., Shen, J., Geng, X.G., Inverse scattering transformation for defocusingN-component NLS with nonzero boundaries. Bull. Malays. Math. Sci. Soc.48, 103,37 (2025)

    2025

  16. [23]

    Liu, H., Shen, J., Geng, X.G., Inverse scattering transformation for theN-component focusing nonlinear Schr¨ odinger equation with nonzero boundary conditions. Lett. Math. Phys.113, 1-47 (2023)

    2023

  17. [24]

    Prinari, B., Inverse scattering transform for nonlinear Schr¨ odinger systems on a nontrivial background: a survey of classical results, new developments and future directions. J. Nonlinear Math. Phys.30, 317-383 (2023)

    2023

  18. [25]

    Asymptotics for the MKdV equation

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

    1993

  19. [26]

    In:Fokas, A.S., Zakharov, V.E

    Deift, P.A., Its, A.R., Zhou,X., Long-time asymptotics for integrable nonlinear wave equations. In:Fokas, A.S., Zakharov, V.E. (eds.) Important Developments in Soliton Theory. Springer Series in Nonlinear Dynamics, Springer, Berlin (1993)

    1993

  20. [27]

    Higher order theory

    Deift, P., Zhou, X., Long-time asymptotics for integrable systems. Higher order theory. Com- mun. Math. Phys.165, 175–191 (1994)

    1994

  21. [28]

    Deift, P., Zhou, X., Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space. Commun. Pure Appl. Math.56, 1029–1077 (2003)

    2003

  22. [29]

    New series: lectures in mathematical sciences, vol

    Deift,P., Zhou,X., Long-time behavior of the non-focusing nonlinear Schr¨ odinger equation, a case study. New series: lectures in mathematical sciences, vol. 5. University of Tokyo, Tokyo (1994)

    1994

  23. [30]

    arXiv:0805.2807

    Dieng, M., McLaughlin, K.D.T.-R., Long-time asymptotics for the NLS equation via ¯∂methods. arXiv:0805.2807

    arXiv

  24. [31]

    Borghese, M., Jenkins, R., McLaughlin, K.T.R., Long time asymptotic behavior of the focusing nonlinear Schr¨ odinger equation. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire.35, 887-920 (2018)

    2018

  25. [32]

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

    2017

  26. [33]

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

    2021

  27. [34]

    Bilman, D., Miller, P.D., A robust inverse scattering transform for the focusing nonlinear Schr¨ odinger equation. Comm. Pure Appl. Math.8, 1722-1805 (2019). 65

    2019

  28. [35]

    Bilman, D., Miller, P.D., Extreme superposition: high-order fundamental rogue waves in the far-field regime. Mem. Amer. Math. Soc.300, 1505 (2024)

    2024

  29. [36]

    Duke Math

    Bilman, D., Ling, L.M., Miller, P.D., Extreme superposition: rogue waves of infinite order and the Painlev´ e-III hierarchy. Duke Math. J.169, 671–760 (2020)

    2020

  30. [37]

    Vartanian, A.H., Long-time asymptotics of solutions to the Cauchy problem for the defocusing non-linear Schr¨ odinger equation with finite-density initial data. I. Solitonless sector. In: Recent developments in integrable systems and Riemann–Hilbert problems (Birmingham, AL, 2...

    2000

  31. [38]

    Vartanian, A.H., Long-time asymptotics of solutions to the Cauchy problem for the defocusing non-linear Schr¨ odinger equation with finite-density initial data. II. Dark solitons on continua. Math. Phys. Anal. Geom.5, 319–413 (2002)

    2002

  32. [39]

    Vartanian, A.H., Exponentially small asymptotics of solutions to the defocusing nonlinear Schr¨ odinger equation. Appl. Math. Lett.16, 425–434 (2003)

    2003

  33. [40]

    Cuccagna, S., Jenkins, R., On the asymptotic stability of soliton solutions of the defocusing nonlinear Schr¨ odinger equation. Commun. Math. Phys.343, 921–969(2016)

    2016

  34. [41]

    Wang, Z.Y., Fan, E.G., Defocusing NLS equation with nonzero background: large-time asymp- totics in a solitonless region. J. Differential Equations.336, 334-373 (2022)

    2022

  35. [42]

    Wang, Z.Y., Fan, E.G., The defocusing nonlinear Schr¨ odinger equation with a nonzero back- ground: Painlev´ e asymptotics in two transition regions. Comm. Math. Phys402, 2879–2930 (2023)

    2023

  36. [43]

    Pure Appl

    Boutet de Monvel, A., Karpenko, I., Shepelsky, D., The modified Camassa–Holm equation on a nonzero background: large-time asymptotics for the Cauchy problem. Pure Appl. Funct. Anal. 7,3 (2022)

    2022

  37. [44]

    arXiv:2412.19703

    Gkogkou, A., Prinari, B., Trogdon, T., Numerical inverse scattering transform for the defocus- ing nonlinear Schr¨ odinger equation with box-type initial conditions on a nonzero background. arXiv:2412.19703

    arXiv

  38. [45]

    Lenells, The nonlinear steepest descent method for Riemann–Hilbert problems of low regu- larity

    J. Lenells, The nonlinear steepest descent method for Riemann–Hilbert problems of low regu- larity. Indiana Univ. Math. J.66, 287–1332 (2017)

    2017

  39. [46]

    Nonlinearity26, 2081–2107 (2013)

    Boutet de Monvel, A., Karpenko, I., Shepelsky, D., A Riemann-Hilbert approach for the Degasperis-Procesi equation. Nonlinearity26, 2081–2107 (2013)

  40. [47]

    Analysis & PDE16, 1351–1388 (2023)

    Charlier, C., Lenells, J., Wang, D.S., The “good” Boussinesq equation: long-time asymptotics. Analysis & PDE16, 1351–1388 (2023)

    2023

  41. [48]

    Yang, Y.L., Fan, E.G., Soliton resolution and large time behavior of solutions to the Cauchy problem for the Novikov equation with a nonzero background. Adv. Math.426109088, 86 (2023)

    2023

  42. [49]

    Geng, X.G., Liu, H., The nonlinear steepest descent method to long-time asymptotics of the coupled nonlinear Schr¨ odinger equation. J. Nonlinear Sci.28, 739–763 (2018)

    2018

  43. [51]

    D., Wang, L., The coupled Hirota equations with a 3×3 Lax pair: Painlev´ e-type asymptotics in transition zone

    Zhao, X. D., Wang, L., The coupled Hirota equations with a 3×3 Lax pair: Painlev´ e-type asymptotics in transition zone. Stud. Appl. Math.153, e12745 (2024)

    2024

  44. [52]

    Liu, N., Zhao, X.D., Guo, B.L., Long-time asymptotic behavior for the matrix modified Korteweg–de Vries equation. Phys. D443, 133560 (2023)

    2023

  45. [53]

    The Cauchy problem of vector complex modified Korteweg–deVries equation: large-time asymptotics with decaying initial data. J. Math. Phys.66, 053506 (2025)

    2025

  46. [54]

    SSIAM, Philadelphia (2016)

    Trogdon T., Olver, S., Riemann–Hilbert problems, their numerical solution, and the computa- tion of nonlinear special functions. SSIAM, Philadelphia (2016)

    2016

  47. [55]

    arXiv:2204.02365

    Charlier, C., Lenells, J., On Boussinesq’s equation for water waves. arXiv:2204.02365

    arXiv

  48. [56]

    Zakharov, A

    V. Zakharov, A. Shabat, A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem: I, Func. Anal. Appl.,8226–235 (1974)

    1974

  49. [57]

    Charlier, C., Lenells, J., The soliton resolution conjecture for the Boussinesq equation. J. Math. Pures Appl.191103621 (2024)

    2024

  50. [58]

    East Asian J

    Abeya, A., Biondini, G., Prinari, B., Inverse scattering transform for the defocusing Manakov system with non-parallel boundary conditions at infinity. East Asian J. Appl. Math.12715–760 (2022)

    2022

  51. [59]

    Physica D368,22–49 (2018)

    Prinari, B., Demontis, F., Li, S., Horikis, T.P., Inverse scattering transform and soliton solu- tions for a square matrix nonlinear Schr¨ odinger equation with nonzero boundary conditions. Physica D368,22–49 (2018)

    2018

  52. [60]

    Xiao, Y., Fan, E.G., Liu, P., Inverse scattering transform for the coupled modified Korteweg-de Vries equation with nonzero boundary conditions. J. Math. Anal. Appl.504, 125567, 31 (2021)

    2021

  53. [61]

    Physica D471, 134434 (2025)

    Han, P.F., Ma, W.X., Ye, R.S., Zhang, Y., Inverse scattering transform for the defocusing- defocusing coupled Hirota equations with non-zero boundary conditions: multiple double-pole solutions. Physica D471, 134434 (2025)

    2025

  54. [62]

    Ma, X.X., Zhu, J.Y., Inverse scattering transform for the two-component Gerdjikov–Ivanov equation with nonzero boundary conditions: Dark-dark solitons. Stud. Appl. Math.151, 676- 715 (2023). 67

    2023

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