Pith. sign in

REVIEW 2 major objections 7 minor 1 cited by

Ground states on a fractured strip and one dimensional reduction

T0 review · 2 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read This paper proves that on a narrow strip with an attractive delta defect, the energy ground state at mass proportional to the width is exactly the one-dimensional delta soliton, constant in the transverse direction.

desk verdict A genuinely new dimensional-reduction result for a delta-defect strip, but the main rigidity theorem is not fully proved as written; the repulsive case additionally rests on an omitted lemma. read the letter →

arxiv 2411.18187 v1 pith:APSAMLMU submitted 2024-11-27 math.AP

Stefan Le Coz (IMT) , Boris Shakarov (IMT) This is my paper · ORCID
classification math.AP MSC 35Q5535A1535B38
keywords nonlinearSchrödingerequationstandingwavesactiongroundstateenergyquantumgraphsfracturedstripdimensionalreduction
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

This paper studies the nonlinear Schrödinger equation on a strip with Neumann boundary conditions and a delta-shaped defect along its centerline, and asks when the two-dimensional ground state is really one-dimensional. For an attractive defect the authors prove full existence of positive action and energy ground states — minimizers of the action on the Nehari manifold, and minimizers of the energy at fixed mass — and then show that as the strip width shrinks to zero the energy minimizer with mass proportional to the width becomes independent of the transverse coordinate: it is exactly the one-dimensional delta soliton extended constantly in y. They also prove that for large widths the ground state must genuinely depend on y, so the transition from one-dimensional to two-dimensional behavior is not just formal. The interest is that this gives a rigorous dimensional-reduction theorem for a nonlinear model with a point defect, the kind of reduction usually taken as an ansatz in quantum-graph and waveguide modelling.

What carries the argument

The central object is the rescaled energy on the fixed-width strip $S=\mathbb{R}\times[0,1]$, $\widetilde{E}_{L,\gamma}(u)=\int_0^1\int_{\mathbb{R}}\big(\frac12|\partial_x u|^2+\frac{1}{2L^2}|\partial_y u|^2-\frac{1}{p+1}|u|^{p+1}\big)dx\,dy+\frac{\gamma}{2}\int_0^1|u(0,y)|^2dy$, whose $1/L^2$ penalty on the transverse derivative makes the dimensional reduction quantitative. The argument uses the explicit one-dimensional delta soliton $\varphi_{\omega,\gamma}(x)=\big(\frac{p+1}{2}\omega\,\mathrm{sech}^2(\frac{p-1}{2}\sqrt{\omega}\,|x|-\tanh^{-1}(\frac{\gamma}{2\sqrt{\omega}}))\big)^{1/(p-1)}$ as the limiting profile, virial (Pohozaev) identities to identify the limiting frequency, and a rigidity lemma: pairing the equation with $-\partial_{yy}u$ yields a coercive quadratic form plus terms that converge to zero, so for small $L$ the transverse derivative must vanish identically.

What would settle it

Numerically minimize the action over the Nehari manifold on the symmetric space for $\gamma=0$ and a sequence of widths $L\to0$; if the minimizer's transverse gradient stays nonzero for arbitrarily small $L$, Lemma 3.16 is false and the small-$L$ branch of Theorem 1.3 collapses. Equivalently, compute $\|\partial_y u_L\|_{L^2}$ for attractive energy minimizers at mass $\tilde m L$: Theorem 1.4 forces it to be exactly zero for all $L<L_*$, so any strictly positive computed value would disprove the rigidity claim.

Watch

Extended reading notes

Core claim

Working on $S_L=\mathbb{R}\times[0,L]$ with $-\partial_{xx}u-\partial_{yy}u+\omega u+\gamma\delta_0(x)u-|u|^{p-1}u=0$ and Neumann conditions, the paper establishes that for $\gamma<0$ and $1<p<3$, for every transverse mass density $\tilde m>0$ there is a critical width $L_*=L_*(\tilde m)$ such that for all $0<L<L_*$ the energy minimizer with mass $m=\tilde m L$ is a function of $x$ alone. This minimizer coincides with the unique positive one-dimensional profile $\varphi_{\omega,\gamma}$ (the explicit sech-type solution of the delta-perturbed line equation) extended constantly in $y$, with frequency $\omega$ determined by $M^{1D}(\varphi_{\omega,\gamma})=\tilde m$. Conversely, there is a second threshold $L_{**}$ such that for $L>L_{**}$ every energy minimizer with the same mass scaling has nontrivial $y$-dependence. The proof passes through a normalized fixed-width problem, establishes convergence of its minimizers to the extended one-dimensional soliton as $L\to 0$, and then uses a rigidity identity (the duality product of the equation with $-\partial_{yy}u$) to force $\partial_y u=0$ below the threshold.

Load-bearing premise

The repulsive-case existence theorem rests on a lemma stated without proof: for a strip with no defect, a sufficiently small width makes the symmetric variational ground state a constant-in-y one-dimensional profile; if that lemma is false, the small-width existence branch in the repulsive case has no foundation.

Editorial extensions

If this is right

  • For every fixed transverse mass density $\tilde m>0$ in the attractive case, there is a critical width $L_*$ below which the energy ground state with mass $\tilde m L$ is exactly the one-dimensional delta soliton extended constantly across the strip, so the 2D model reduces rigorously to the 1D delta model.
  • Above a larger threshold $L_{**}$ the same mass scaling forces the minimizer to have genuine transverse dependence, so the reduction to 1D cannot hold uniformly in the width.
  • The normalized energies converge: $L^{-1} e_{\tilde m L,\gamma} \to e^{1D}_{1,\gamma}$ as $L\to 0$, giving a quantitative sense in which the 1D model captures the ground-state energy of the thin strip.
  • In the repulsive case, symmetric action ground states exist for sufficiently small defect strength or small width, despite the run-away instability that prevents unconstrained minimizers.

Reading between the lines

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

  • Beyond the paper: the rigidity pairing with $-\partial_{yy}u$ is a general mechanism; the same argument should yield dimensional reduction for thin domains with other transverse geometries and Neumann structure, not just rectangular strips.
  • Beyond the paper: the two thresholds $L_*$ and $L_{**}$ suggest a critical width at which transverse modulation bifurcates from the constant profile; numerical continuation could map this curve in $(\tilde m,\gamma,p)$ and compare it with the variational bound in Proposition 5.4.
  • Beyond the paper: the missing proof of Lemma 3.16 is likely obtainable by the same rigidity argument used for the attractive case; if so, the small-width reduction would extend to repulsive defects, and if not, the repulsive existence theorem would need a different mechanism.
  • Beyond the paper: the mass scaling $m=\tilde m L$ is essential to the conclusion; fixing the absolute mass instead would let minimizers spread in $x$, so any experimental or numerical test of the 1D-to-2D transition must use the linear mass scaling.
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

2 major / 7 minor

Summary. This paper studies the nonlinear Schrödinger equation on a strip S_L = R × (0,L) with Neumann boundary conditions and a δ-interaction supported on the x-axis. The authors prove existence of action ground states in the attractive case (Theorem 1.1) and, under smallness restrictions on γ or L, in the repulsive case (Theorem 1.3). They also prove existence of energy ground states in the attractive case for 1<p<3 (Theorem 1.2), and show their absence in the repulsive case (Lemma 4.4). The central new results are Theorems 1.4 and 1.5, which assert that, for the attractive case, energy minimizers with mass m̃L are exactly one-dimensional (independent of y) when L is sufficiently small, and genuinely two-dimensional when L is sufficiently large. The one-dimensional limit is identified as the known soliton of the 1D NLS with a δ-potential.

Significance. The existence results for the attractive case are well supported and constitute a useful contribution to the variational theory of NLS with singular potentials on domains with mixed dimensionality. The dimensional reduction statements (Theorems 1.4 and 1.5) are the main advertised achievements and, if fully established, would provide a rigorous bridge between nonlinear quantum graphs and thin-strip models. The paper also gives explicit formulas for the 1D profiles, a careful trace theory, and several qualitative properties (exponential decay, symmetry, monotonicity). The numerical illustrations agree with the theorems. The weaknesses are two load-bearing gaps: an unproved lemma controlling the symmetric action ground state for small L (Lemma 3.16) and an incomplete rigidity argument in Lemma 5.3 that is essential for Theorem 1.4. These issues currently prevent the paper from fully delivering its central claims.

major comments (2)
  1. [Section 3.2, Lemma 3.16] Lemma 3.16 is stated without proof: the text says 'This result can proved following a similar reasoning to the one of Section 5, we omit the details here.' This lemma is load-bearing for Lemma 3.17 and hence for the L<L† part of Theorem 1.3, since it is used to identify s_{ω,0,sym} with the y-independent profile φ_{ω,0}. The reasoning of Section 5 concerns energy minimizers in the attractive case, not action minimizers in the repulsive case, and the adaptation is non-obvious (in particular, the role of the symmetry constraint and the absence of a positivity condition like γ<0). The authors should provide a complete proof of Lemma 3.16 or explicitly state it as a conjecture and remove the L<L† case from Theorem 1.3.
  2. [Section 5, Lemma 5.3] The rigidity argument proving exact one-dimensionality of the minimizers does not work as written. From identity (74), the authors show that the sum of the quadratic form and the (1/L_n^2-1)∥∂_y w_n∥^2 term is positive and that the last line tends to zero, then conclude that w_n = ∂_y u_n = 0. This is a non sequitur: positivity plus a remainder tending to zero only gives ∥∂_y w_n∥ → 0 (and ∥w_n∥ → 0), not w_n ≡ 0. A sequence of y-dependent minimizers could converge to the y-independent limit. To obtain exact vanishing one must prove a uniform quantitative estimate, e.g., that the negative term p∫∫|φ|^{p-1}|w_n|^2 is absorbed by a fixed fraction of the positive quadratic form plus (1/L_n^2-1)∥∂_y w_n∥^2, uniformly for large n. The paper does not supply such an estimate; the sentence 'the sum between the second and the third line is positive' is not justified by the displayed inequalities. Since Lemma 5.3 is the key step in the proof of Theorem 1.4, Theorem 1.4 is not fully established.
minor comments (7)
  1. [Equation (69)] The coefficient of L_n^{-2}∥∂_y u_n∥^2 in (69) appears to be incorrect; direct algebra from (68) gives +2(p-1)/(5-p) rather than -4/(5-p). Since this term vanishes in the limit L_n→0, the proof of Lemma 5.2 goes through, but the displayed formula should be corrected.
  2. [Lemma 3.16 statement] The statement says the minimizer is 'up to translation and phase shift' the profile φ_{ω,0} trivially extended, but the space H^1_sym(S_L) fixes the center at x=0; translations within this space are trivial, so the wording is misleading.
  3. [Lemma 3.15] The notation S_γ in the statement of Lemma 3.15 is undefined; it should be the action functional S_{ω,γ} used throughout the paper.
  4. [Lemma 3.16 proof] The sentence 'This result can proved' is grammatically incorrect; it should read 'can be proved'.
  5. [Lemma 5.1, Eq. (64)] In the line 'that is mL→ 1 in Lq(0, 1) for any q∈ [1,∞) as L→∞', the limit 'L→∞' should be 'L→0'.
  6. [Lemma 3.17] The inequality ∥φ_{ω,γ}∥_{L^{p+1}(S_L)}^{p+1} < 2∥φ_{ω,0}∥_{L^{p+1}(S_L)}^{p+1} is stated without proof; a short justification would be helpful since it is used in the contradiction argument.
  7. [Lemma 4.3] Lemma 4.3 is stated without proof, referring to an adaptation of [1, Lemma 3.3]. Given its central role in Theorem 1.2, a brief indication of the adaptation (e.g., the role of the strip geometry and the inhomogeneous δ-term) is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the 1D delta-soliton inputs are independent external results, and the 2D existence and dimensional-reduction claims are derived rather than assumed.

full rationale

The derivation chain is not circular. The one-dimensional delta-potential theory used as input (explicit profile (14), variational characterization in Proposition 2.3, mass-frequency monotonicity in Lemma 2.6) is imported from the independent references [19,20,29] and [2,6]; although [29] is co-authored by a present author, the same facts are established in other independent works, so the citations are real evidence rather than a self-citation chain. The two-dimensional existence results (Theorems 1.1-1.3) are proved by direct variational arguments: Nehari minimization, coercivity via Lemma 3.5, concentration-compactness, and profile decomposition. The dimensional-reduction theorems (1.4-1.5) are derived by rescaling to a fixed strip, proving convergence of the energy levels (Lemma 5.1), strong H1 convergence to the extended one-dimensional soliton (Lemma 5.2), and then attempting a rigidity argument (Lemma 5.3) to upgrade convergence to exact y-independence. Two admitted gaps are present but they are not circularity. Lemma 3.16 is explicitly stated without proof ('This result can proved following a similar reasoning to the one of Section 5, we omit the details here') and is load-bearing for Theorem 1.3; this is an omitted proof, not the conclusion being assumed. The rigidity step in Lemma 5.3 is not fully justified as written: positivity of the quadratic form plus convergence of the remainder in (74) yields only convergence of ∂_y u_n, and the assertion that 'the sum between the second and the third line is positive for n large enough' is not established by the displayed inequalities without a uniform L-infinity bound on u_n; this is a rigor gap concerning Theorem 1.4, not a circular reduction. No fitted parameter is renamed a prediction, no known result is merely relabeled, and no load-bearing premise reduces to the paper's own conclusion. Score 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. It relies on the established 1-D delta soliton theory and on adaptations of standard concentration-compactness tools to the strip, plus one unproved auxiliary lemma (Lemma 3.16).

assumptions (3)
  • domain assumption One-dimensional delta NLS ground-state theory: explicit profile (14), variational characterization (Proposition 2.3), and mass monotonicity (Lemma 2.6).
    Imported from prior literature [2,6,19,20,29] without reproof; these are standard, explicit results, some co-authored by the present authors.
  • domain assumption Adapted concentration-compactness principle on the strip (Lemma 4.3) and profile decomposition (Lemma 3.15).
    The paper states these are adapted 'mutatis mutandis' from graph and R^d settings [1,26]; no detailed proof is given for the strip adaptation.
  • ad hoc to paper Lemma 3.16: for γ = 0 and small L, s_{ω,0,sym} is achieved by the y-independent 1-D soliton.
    Stated without proof in Section 3.2; load-bearing for Theorem 1.3 (repulsive action ground states for small L).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Ground states on a fractured strip and one dimensional reduction." pith.science (2026). https://pith.science/paper/APSAMLMU

@misc{pith2026241118187,
  author       = {Pith},
  title        = {Pith review of: Ground states on a fractured strip and one dimensional reduction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/APSAMLMU}},
  note         = {Machine review of arXiv:2411.18187}
}
abstract

We consider the nonlinear Schr\''odinger equation on a strip with Neumann boundary conditions and a delta condition on the $x$-axis. First, we show the existence of ground states as minimizers of the action or of the energy under suitable constraints. Second, we prove that the energy minimizers converge to the ground state on the line with a delta condition as the amplitude of the strip shrinks to zero.

Figures

Figures reproduced from arXiv: 2411.18187 by the authors.

Figure 1
Figure 1. Outcomes of numerical minimization over the Nehari manifold when γ = −1 for a strip of length 2.5 (left picture) and length 6 (right picture) 2.1. The trace theorem. In this section, we define rigorously the traces that are used along the work and present the properties needed for the proofs. We start by defining the trace of the functions projected on the hyperplane x = 0. Let us denote by (τf)(x, y) = f(0, y) for … view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Transverse Instability and Bifurcation Analysis of the line soliton for the NLS equation on a Fractured Strip

    math.AP 2026-08 conditional novelty 7.0 of 10

    For NLS on a strip with an attractive delta interaction, line solitons are orbitally stable below a critical width and unstable above, with a pitchfork branch of new positive solutions at the threshold.

Reference graph

Works this paper leans on

35 extracted references · 31 canonical work pages · cited by 1 Pith paper

  1. [1]

    Adami, C

    R. Adami, C. Cacciapuoti, D. Finco, and D. Noja. Constrained energy minimization and orbital stability for the NLS equation on a star graph. Ann. Inst. Henri Poincar´ e, Anal. Non Lin´ eaire, 31(6):1289–1310, 2014

    work page 2014

  2. [2]

    Adami, D

    R. Adami, D. Noja, and N. Visciglia. Constrained energy minimization and ground states for NLS with point defects. Discrete Contin. Dyn. Syst., Ser. B , 18(5):1155–1188, 2013

    work page 2013

  3. [3]

    Akahori, Y

    T. Akahori, Y. Bahri, S. Ibrahim, and H. Kikuchi. Pitchfork bifurcation at line solitons for nonlinear Schr¨ odinger equations on the product spaceR× T. Ann. Henri Poincar´ e, 25(7):3467–3497, 2024

    work page 2024

  4. [4]

    Berestycki and T

    H. Berestycki and T. Cazenave. Instabilit´ e des ´ etats stationnaires dans les ´ equations de Schr¨ odinger et de Klein-Gordon non lin´ eaires.C. R. Acad. Sci. Paris , 293(9):489–492, 1981

    work page 1981

  5. [5]

    Berestycki and T

    H. Berestycki and T. Lachand-Robert. Some properties of monotone rearrangement with applications to elliptic equations in cylinders. Mathematische Nachrichten, 266(1):3–19, 2004

    work page 2004

  6. [6]

    Boni and R

    F. Boni and R. Carlone. NLS ground states on the half-line with point interactions. NoDEA Nonlinear Differential Equations Appl., 30(4):Paper No. 51, 23, 2023

    2023

  7. [7]

    H. Brezis. Functional analysis, Sobolev spaces and partial differential equations , volume 2,3. Springer, 2011

    work page 2011

  8. [8]

    Brezis and E

    H. Brezis and E. Lieb. A relation between pointwise convergence of functions and convergence of func- tionals. Proceedings of the American Mathematical Society, 88(3):486–490, 1983

    work page 1983

Show all 35 references
  1. [9]

    J. E. Brothers and W. P. Ziemer. Minimal rearrangements of Sobolev functions. Acta Univ. Carolin. Math. Phys., 28(2):13–24, 1987. 15th winter school in abstract analysis (Srn´ ı, 1987)

    1987

  2. [10]

    Cazenave

    T. Cazenave. Semilinear Schr¨ odinger Equations, volume 10. American Mathematical Soc., 2003

    2003

  3. [11]

    Cazenave and P.-L

    T. Cazenave and P.-L. Lions. Orbital stability of standing waves for some nonlinear Schr¨ odinger equa- tions. Communications in Mathematical Physics , 85:549–561, 1982

    1982

  4. [12]

    Cianchi and N

    A. Cianchi and N. Fusco. Steiner symmetric extremals in P´ olya–Szeg¨ o-type inequalities. Advances in Mathematics, 203(2):673–728, 2006

    2006

  5. [13]

    De Coster, S

    C. De Coster, S. Dovetta, D. Galant, and E. Serra. On the notion of ground state for nonlinear Schr¨ odinger equations on metric graphs.Calc. Var. Partial Differ. Equ. , 62(5):28, 2023. Id/No 159

    2023

  6. [14]

    de Laire, P

    A. de Laire, P. Gravejat, and D. Smets. Minimizing travelling waves for the Gross-Pitaevskii equation on R× T. Annales de la Facult´ e des Sciences de Toulouse. Math´ ematiques., to appear, 2024

    2024

  7. [15]

    Dovetta, E

    S. Dovetta, E. Serra, and P. Tilli. Action versus energy ground states in nonlinear Schr¨ odinger equations. Math. Ann., 385(3-4):1545–1576, 2023

    2023

  8. [16]

    D. G. Duffy. Green’s functions with applications . Adv. Appl. Math. (Boca Raton). Boca Raton, FL: CRC Press, 2nd ed. edition, 2015

    2015

  9. [17]

    L. C. Evans and R. F. Gariepy. Measure theory and fine properties of functions . Textb. Math. Boca Raton, FL: CRC Press, 2nd revised ed. edition, 2015

    2015

  10. [18]

    Exner and O

    P. Exner and O. Post. Quantum networks modeled by graphs. AIP Conference Proceedings, 998(1):1–17, 2008

    2008

  11. [19]

    Fukuizumi and L

    R. Fukuizumi and L. Jeanjean. Stability of standing waves for a nonlinear Schrodinger equation with a repulsive Dirac delta potential. Discrete and Continuous Dynamical Systems , 21(1):121, 2008

    2008

  12. [20]

    Fukuizumi, M

    R. Fukuizumi, M. Ohta, and T. Ozawa. Nonlinear schr¨ odinger equation with a point defect. InAnnales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire, volume 25,5, pages 837–845. Elsevier, 2008

    2008

  13. [21]

    Genoud, S

    F. Genoud, S. Le Coz, and J. Royer. A minimal mass blow-up solution on a nonlinear quantum star graph. arXiv:2302.09678, 2023

    2023 arXiv

  14. [22]

    R. H. Goodman, P. J. Holmes, and M. I. Weinstein. Strong NLS soliton–defect interactions. Phys. D: Nonlinear Phenomena, 192(3-4):215–248, 2004

    2004

  15. [23]

    Grisvard

    P. Grisvard. Elliptic problems in nonsmooth domains , volume 69 of Class. Appl. Math. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), reprint of the 1985 hardback ed. edition, 2011

    1985

  16. [24]

    Gustafson and T

    S. Gustafson and T. Inui. Threshold even solutions to the nonlinear Schr¨ odinger equation with delta potential at high frequencies. Discrete Contin. Dyn. Syst. , 44(10):3135–3176, 2024

    2024

  17. [25]

    Jeanjean and S.-S

    L. Jeanjean and S.-S. Lu. On global minimizers for a mass-constrained problem. Calc. Var. Partial Differ. Equ., 61(6):18, 2022. 34 S. LE COZ AND B. SHAKAROV

    2022

  18. [26]

    Jeanjean and K

    L. Jeanjean and K. Tanaka. A positive solution for a nonlinear Schr¨ odinger equation on RN. Indiana Univ. Math. J. , 54(2):443–464, 2005

    2005

  19. [27]

    S. Kosugi. A semilinear elliptic equation in a thin network-shaped domain. Journal of the Mathematical Society of Japan, 52(3):673–697, 2000

    2000

  20. [28]

    S. Kosugi. Semilinear elliptic equations on thin network-shaped domains with variable thickness. J. Differ. Equations, 183(1):165–188, 2002

    2002

  21. [29]

    Le Coz, R

    S. Le Coz, R. Fukuizumi, G. Fibich, B. Ksherim, and Y. Sivan. Instability of bound states of a nonlinear Schr¨ odinger equation with a Dirac potential.Phys. D, 237(8):1103–1128, 2008

    2008

  22. [30]

    E. H. Lieb. Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation. Studies in Applied Mathematics , 57(2):93–105, 1977

    1977

  23. [31]

    Mari¸ s and A

    M. Mari¸ s and A. Mur. Periodic traveling waves for nonlinear schr¨ odinger equations with non-zero con- ditions at infinity in R2. arXiv:2404.11772, 2024

    2024 arXiv

  24. [32]

    O. Post. Spectral analysis on graph-like spaces, volume 2039 of Lect. Notes Math. Berlin: Springer, 2012

    2012

  25. [33]

    Terracini, N

    S. Terracini, N. Tzvetkov, and N. Visciglia. The nonlinear Schr¨ odinger equation ground states on product spaces. Analysis & PDE , 7(1):73–96, 2014

    2014

  26. [34]

    M. Willem. Minimax theorems, volume 24 of Prog. Nonlinear Differ. Equ. Appl. Boston: Birkh¨ auser, 1996

    1996

  27. [35]

    Yamazaki

    Y. Yamazaki. Stability of line standing waves near the bifurcation point for nonlinear Schr¨ odinger equations. Kodai Math. J., 38(1):65–96, 2015. Institut de Math´ematiques de Toulouse ; UMR5219, Universit´e de Toulouse ; CNRS, UPS IMT, F-31062 Toulouse Cedex 9 (France) Email...

    2015

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.