Mode stability for self-similar blowup of slightly supercritical NLS: I. low-energy spectrum

Zexing Li

REVIEW 3 major objections 4 minor 51 references

Mode stability for self-similar blowup of slightly supercritical NLS: I. low-energy spectrum

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read For slightly supercritical NLS, the linearized operator around the self-similar blowup profile has low-energy discrete spectrum exactly {0, −bi, −2bi}, so the only low-energy unstable modes are the symmetry-generated ones.

desk verdict A careful, substantial low-energy mode stability proof for self-similar NLS blowup; the flagged Definition 4.15 inconsistency looks like a typesetting garble rather than a real gap. read the letter →

arxiv 2507.11248 v1 pith:R6RMEPKN submitted 2025-07-15 math.AP math.SP

Zexing Li This is my paper · ORCID

classification math.APmath.SP MSC 35Q5535B4035P15

keywords modestabilityself-similarblowupslightlysupercriticalNLSlinearizedoperatorJostfunctionspectralbifurcationWKBanalysiseigenvalueasymptotics

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 proves mode stability for the linearized operator around the self-similar blowup profile in the nonlinear Schrödinger equation just above the mass-critical regime ($0 < s_c \ll 1$). Mode stability means that every unstable direction of the linearized flow is a symmetry mode — phase rotation, translation, or scaling — so that those directions can be removed by choosing the right reference frame. The theorem states that, for small enough $s_c$ and for wavelengths $\sigma$ just above $s_c$, the discrete spectrum of $H_b$ in the low-energy region $\{\Im z < b(\sigma - s_c), |z| \leq \delta\}$ is exactly $\{0, -bi, -2bi\}$, with the Riesz projections of dimension 1, $d$, and 1 respectively. If true, this removes the spectral obstruction to asymptotic stability of self-similar blowup and shows that no quantized unstable eigenvalue bifurcates from the known nullspace of the ground-state linearized operator.

What carries the argument

The object that carries the argument is $H_b$, the linearized operator obtained by writing the renormalized NLS flow around the self-similar profile $Q_b$. $H_b$ is a non-self-adjoint $2\times 2$ Schrödinger-type system with a coupling $-ib\Lambda_0$ to the scaling generator; it preserves spherical harmonics, so the spectral problem is studied class by class. The proof builds, for each class, local fundamental solutions from scalar approximate operators: modified Bessel functions near zero, a quadratic WKB problem with C-valued Airy functions near the turning point $r \approx 2/b$, and free solutions in the high-angular-momentum regime where the centrifugal term dominates. These are matched across the domains and used to define the Jost function, an analytic function of the spectral parameter whose zeros, with multiplicity, are exactly the eigenvalues. Continuity of the Jost function in $b$ at $b = 0$ transfers the known generalized nullspace of the ground-state operator into a uniqueness statement for the bifurcating eigenvalues.

What would settle it

Numerically compute the low-energy spectrum of $H_b$ for several small $b$ in $d = 1$ and $d = 2$: the theorem says that in $\{|z| \leq \delta, \Im z < b(\sigma - s_c)\}$ the only eigenvalues are $0$, $-bi$ (multiplicity $d$), and $-2bi$. A direct check of the predicted leading order $\lambda_{3,b} = \frac{4\pi\kappa_Q^2}{\int_0^\infty Q^2 r^{d+1}\,dr}\, b^{-3} e^{-\pi/b}$ at moderately small $b$, for instance by solving the radial eigenvalue problem with a spectral method and extrapolating in $b^{-3}e^{-\pi/b}$, would settle whether the bifurcated mode actually sits where the paper places it.

Watch

Extended reading notes

Core claim

The central claim of Theorem 1.1 is that the low-energy spectrum of the matrix linearized operator $H_b$ is completely accounted for by the three continuous symmetries of the equation. In each spherical class the proof constructs admissible fundamental solutions and shows that their matching Wronskian, the Jost function, has prescribed zeros. In the radial class the two bifurcated eigenvalues are $\lambda_{2,b} = 2b(1 + o_{s_c\to 0}(1))$ and $\lambda_{3,b} = \frac{4\pi\kappa_Q^2}{\int_0^\infty Q^2 r^{d+1}\,dr}\, b^{-3} e^{-\pi/b}(1+o(1))$; in the first spherical class the eigenvalue $-bi$ has multiplicity $d$. No other eigenvalues appear in the low-energy window, and none of the listed eigenvalues has generalized eigenfunctions. The result is a spectral bifurcation statement: as $s_c \to 0$ (equivalently $b \to 0$), the spectrum degenerates toward that of the ground-state linearized operator, and the theorem identifies exactly which pieces of the generalized nullspace survive as true eigenvalues.

Load-bearing premise

The load-bearing premise is the imported quantitative asymptotics of the self-similar profile $Q_b$, including $s_c \sim b^{-1}e^{-\pi/b}$ and the pointwise bounds on $Q_b$ and its derivatives; the paper does not re-derive these estimates, and if they failed the spectral conclusion would collapse.

Editorial extensions

If this is right

The low-energy region contains no unstable modes other than the three symmetry families, so any instability in the nonlinear evolution must either be high-energy or come from the symmetry parameters themselves.
The explicit asymptotics $\lambda_{2,b} \approx 2b$ and $\lambda_{3,b} \approx \mathrm{const}\cdot b^{-3}e^{-\pi/b}$ give concrete rates at which the bifurcated modes cross the imaginary axis.
The non-existence of generalized eigenfunctions means the symmetry modes do not excite Jordan chains inside the low-energy window.
For high spherical classes the matching of admissible branches fails, so eigenvalues cannot appear for large angular momentum in the window considered; extending the bound from $\Im\lambda < b/2$ to fixed $I_0 b$ is stated as possible with additional effort.
The paper states that the complete mode stability result, including the rest of the spectrum, is left to a companion paper; the low-energy theorem supplies the part needed for counting the unstable directions.

Reading between the lines

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

Beyond the paper: the same Jost-function plus matched-WKB route should transfer to other slightly supercritical dispersive models, such as the quartic gKdV whose self-similar profiles were constructed by the same profile method.
Beyond the paper: a direct numerical computation of the spectrum of $H_b$ in $d = 1$ and $d = 2$ for $b = 10^{-2}$ and $10^{-3}$ could test the emptiness of the low-energy window and the leading-order formula for $\lambda_{3,b}$; the predicted eigenvalue is exponentially small in $1/b$, so high precision is needed.
Beyond the paper: the theorem's reliance on imported profile asymptotics suggests that future improvements of those estimates would sharpen the allowed range of $s_c$, while the proof's structure suggests the window $\{\Im z < b(\sigma-s_c)\}$ can be enlarged to $I_0 b$ by strengthening the exterior inversion estimates, as the paper remarks.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper proves low-energy mode stability for the linearized operator around the self-similar profile in slightly mass-supercritical NLS, in all dimensions d ≥ 1. That is genuinely new: it pushes Perelman's Jost-function argument and the Bahri–Martel–Raphaël WKB construction into a non-self-adjoint, matrix-valued, high-dimensional setting. The main theorem is clearly stated, with the dimensions of the three Riesz projections spelled out, and the proof is organized as a sequence of explicit scalar approximation regimes, inversion lemmas, and matching steps. The paper gives real credit where it is due: Proposition 2.4 imports the profile asymptotics from [1] openly, and the later sections refine rather than hide those bounds. That is a legitimate division of labor, not a hidden assumption.

The stress-test concern about Definition 4.15 does not survive a close reading. The displayed formula in the text as I received it is garbled: it seems to say μ^4 = b^2(4ν^2−1)/(16E^2) while also defining μ = e^{iπ/4}(2E/b)^{1/2}. Those two statements conflict. But the surrounding mathematics pins down the intended meaning. A direct change of variables in (4.134)–(4.136) forces α = b^2(4ν^2−1)/(16E^2), which is positive with arg α = −2 arg E, exactly as Remark 4.16(1) and Lemmas 4.17 and 4.19 require. And the Airy reduction needs the usual μ^4 = −4E^2/b^2, which is consistent with the displayed μ. So the formula line appears to be a typo or OCR artifact in the equation, not a mistake in the argument. The author should fix the notation, but the high-spherical-class WKB construction is not unsound on this ground.

Two softer concerns are worth naming. First, the final theorem is conditional on the imported profile estimates of Proposition 2.4; the paper says this explicitly, and it is a frontier dependency rather than a circularity, but it does mean the spectral conclusion inherits any error in [1]. Second, the final Jost-function uniqueness argument in Section 7 is only partially visible in the reviewed excerpt; the visible setup and the continuity estimates are consistent, but independent verification of the full proof will be heavy.

This is a paper for specialists in NLS blowup and spectral stability. It deserves a serious referee. I would recommend sending it to peer review; the central claim looks well supported, and the main required change is a careful proofreading pass on Definition 4.15 and the surrounding notational clutter.

Referee Report

3 major / 4 minor

Summary. The paper studies the linearized operator H_b around the self-similar profile Q_b for slightly mass-supercritical NLS and proves low-energy mode stability in d ≥ 1. The main theorem (Theorem 1.1) asserts that, for s_c > 0 sufficiently small and for the profile constructed by Bahri-Martel-Raphaël, the discrete spectrum of H_b in the region {ℑ z < b(σ − s_c), |z| ≤ δ} consists exactly of {0, −bi, −2bi}, with Riesz projections of dimensions 1, d, and 1. The proof reduces the eigenvalue problem to a radial ODE system, constructs admissible interior and exterior fundamental solutions using scalar WKB approximations and special functions, matches them at intermediate points, builds the two bifurcated radial eigenmodes, and then uses a Jost-function uniqueness argument. The high-spherical-class part of the proof is based on a separate WKB construction with a parameter α. The paper is long and carefully structured, with detailed lemmas and estimates, but one load-bearing definition in the high-spherical-class section is internally inconsistent as written.

Significance. If Theorem 1.1 is correct, it supplies the missing spectral ingredient in the route to asymptotic stability of self-similar blowup for slightly supercritical NLS, and it gives a sharp count of low-energy unstable modes. The methodology is also of independent interest: the Jost-function argument for non-self-adjoint operators, the quantitative C-valued WKB analysis near turning points, and the uniform-in-ν estimates for high spherical classes are substantive technical contributions. The paper does not engage in any parameter fitting or definitional circularity: the eigenvalues are constructed explicitly and counted through independent matching and Jost-function arguments. The main caveat is that the proof imports the full quantitative profile asymptotics from [1] as Proposition 2.4, so the spectral conclusion is conditional on those estimates; this is a dependency rather than a flaw. Subject to the correction of the Definition 4.15 inconsistency discussed below, the central claim is credible and the manuscript is a significant contribution.

major comments (3)

[§4.2, Definition 4.15 and Eq. (4.137)] The displayed definitions of α and μ^4 are internally inconsistent and inconsistent with the rest of the section. A direct change of variables in Eq. (4.134), namely s = br/(2√E), gives the parameter α = b^2(ν^2−1/4)/(4E^2), not the displayed α = −(ν^2−1/4)/4. Moreover, the displayed μ = e^{iπ/4}(2E/b)^{1/2} gives μ^4 = −4E^2/b^2, not the displayed b^2(4ν^2−1)/(16E^2). Remark 4.16(1) itself states that arg α = −2 arg E, which holds only for the corrected positive α. Because Lemma 4.17, Lemma 4.19, Proposition 4.20, Proposition 5.2, and the high-spherical-class non-existence argument in Section 7.2 all rely on α ∈ C_{θ0} and on the WKB asymptotics built from these definitions, the high-ν branch of the proof is unsound as written. The intended correction is local and identifiable, but the author must explicitly fix the definition of α, reconcile the displayed μ^4 with the stated μ (likely μ^4 = −4E^2/b^2), and re-verify the estimates that depend on it, in particular (4.144)–(4.146), (4.152)–(4.158), (4.168), and (4.171).
[§5.2 and §7.2] The exclusion of eigenvalues in high spherical classes, which is required for Theorem 1.1's claim that no non-symmetry low-energy eigenvalues exist, rests on the boundary asymptotics (5.139)–(5.140) at x_* = b^{−1/2}. Those asymptotics are obtained from the high-ν WKB solutions of Definition 4.15 and from the inversion bounds of Lemma 4.21. Therefore the inconsistency in Definition 4.15 propagates into Proposition 5.2 and into the final non-existence argument in Section 7.2. A repair of the definition alone is not sufficient; the author should also confirm that the corrected parameter α satisfies the estimates used in Lemma 4.21 and in the mismatch argument at x_*, and should state explicitly where each of (5.139)–(5.140) is re-verified.
[§4.2.1, Remark 4.16(1)] Remark 4.16(1) refers to '(4.12)' when defining the analytic function ζ, but the relevant equation in the high-spherical-class section is (4.136), not (4.12). This is a citation error that obscures the definitional issue in Definition 4.15; it should be corrected together with the definition itself.

minor comments (4)

[§5.2, opening sentence] The sentence 'there exist four smooth functions Φ_{j;b,λ,ν} for j = 1, 2' is internally inconsistent: only two functions are listed and only two admissible branches are constructed. The wording should say 'two smooth functions' or 'two vector-valued fundamental solutions'.
[§1.2, Theorem 1.1] The parameter σ appears in the condition 0 < σ − s_c ≤ ε^*(s_c) but is not introduced in the theorem statement; the reader must infer from [31, Proposition 4.5] that σ is the Sobolev exponent in (˙H^σ(R^d))^2. The author should state this explicitly for self-containedness.
[§4.2, Definition 4.15] Even after the intended correction, the manuscript would benefit from an explicit formula for μ^4 in terms of E and b, so that the relation between μ, μ^{4/3}, and the WKB phase η is unambiguous; the current text leaves the reader to reconcile the displayed μ^4 with the displayed μ.
[§4.1.1, Proposition 4.8(4)] The statement that 'ψ^{b,1}_1 is non-vanishing on R' should presumably read 'on (0, ∞)' or 'on r ≥ 0', since the WKB solution is constructed for r > 0; this is a minor wording issue.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spectral conclusion is obtained from external profile asymptotics and an independent Jost-function/WKB analysis, not from its own statement.

full rationale

The claimed derivation chain does not reduce to its inputs. Theorem 1.1 concerns the discrete low-energy spectrum of Hb, and its proof depends on external quantitative information about the self-similar profile Qb imported from Bahri–Martel–Raphaël [1] as Proposition 2.4 (estimates (2.9)–(2.16)). That imported theorem supplies existence and asymptotics of the profile, but it does not contain, name, or otherwise determine the target spectral set {0, −bi, −2bi}; the spectral count is obtained by explicitly constructing fundamental solutions, matching interior and exterior asymptotic branches, and proving uniqueness of zeros of the Jost function (Lemmas 7.1, 7.2 and 7.4). The bifurcated eigenmodes are located by a self-contained matching computation in Section 6, not fitted to the desired conclusion. The only self-citation, [31], is used for the prior reduction of the nonlinear asymptotic-stability problem to linear mode stability and for the location of the essential spectrum; it is not used to define the discrete eigenvalue set or to forbid alternative eigenvalues, and the uniqueness argument is proved in the present paper rather than imported. No equation is defined in terms of the predicted spectrum, no fitted parameter is renamed as a prediction, and no load-bearing claim rests on a self-citation chain. The internal inconsistency flagged in Definition 4.15 concerning the sign/scaling of α is a potential correctness issue in the high-spherical-class WKB construction, but it is not a circularity: even if that construction must be corrected, the theorem's conclusion is not assumed as an input. The derivation is therefore self-contained from the stated external profile theorem onward.

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

The paper introduces no free parameters fitted to data and no new physical entities. The bifurcation parameter b is taken from the profile construction in [1], not chosen to fit the spectral conclusion. The main axioms are standard mathematical facts, the imported profile construction, and the previously established reduction from nonlinear stability to linear mode stability. These are honest external premises, not hidden assumptions tailored to force the theorem.

assumptions (5)

standard math Existence, uniqueness, positivity, and exponential asymptotics of the ground state Q of mass-critical NLS.
Used in Section 2.1 and throughout; cited to [46, 47, 6], with statements in Lemma 2.1.
standard math The generalized kernel of H0 is exactly (2d+4)-dimensional, spanned by the vectors in (2.7).
Used as the b=0 base for the bifurcation argument and for the Jost function uniqueness at b=0; cited to [49, 27, 7] in Proposition 2.3.
domain assumption The self-similar profile Q_b exists with the quantitative asymptotics in Proposition 2.4, including the relation s_c ∼ b^{-1} e^{-π/b}.
Imported from Bahri-Martel-Raphaël [1] and used as the starting point for all exterior WKB constructions and matching asymptotics.
domain assumption The nonlinear asymptotic stability problem reduces to linear mode stability of H_b, as established in [31].
Motivates the spectral problem and places Theorem 1.1 as the missing linear piece; the reduction is not reproved here.
standard math Standard properties of modified Bessel functions and Airy functions used in Sections 3-4.
Used for fundamental solutions and WKB approximations; cited to standard references and proved in part in Lemmas 3.3 and 4.1.

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Cite this review

Pith. "Pith review of Mode stability for self-similar blowup of slightly supercritical NLS: I. low-energy spectrum." pith.science (2026). https://pith.science/paper/R6RMEPKN

@misc{pith2026250711248,
  author       = {Pith},
  title        = {Pith review of: Mode stability for self-similar blowup of slightly supercritical NLS: I. low-energy spectrum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R6RMEPKN}},
  note         = {Machine review of arXiv:2507.11248}
}

abstract

We consider self-similar blowup for (NLS) $i\partial_t u + \Delta u + u|u|^{p-1} = 0$ in $d \ge 1$ and slightly mass-supercritical range $0 < s_c := \frac d2 - \frac{2}{p-1} \ll 1$. The existence and stability of such dynamics [Merle-Rapha\"el-Szeftel, 2010] and construction of suitable profiles [Bahri-Martel-Rapha\"el, 2021] lead to the question of asymptotic stability. Based on our previous work [Li, 2023], this nonlinear problem is reduced to linear mode stability of the matrix linearized operator. In this work, we prove mode stability for the low-energy spectrum in $d \ge 1$ as a perturbation of the linearized operator around ground state for mass-critical NLS. The main difficulty of this spectral bifurcation problem arises from the non-self-adjoint, relatively unbounded and high-dimensional nature, for which we exploit the Jost function argument from [Perelman, 2001], qualitative WKB analysis generalized from [Bahri-Martel-Rapha\"el, 2021], matched asymptotics method and uniform estimates for high spherical classes based on special functions.

Figures

Figures reproduced from arXiv: 2507.11248 by the authors.

**Figure 1.** Spectrum of H0 and Hb near the origin: red line for σess, blue cross for eigenpairs in radial class and green circle for eigenpairs in first spherical classes. In particular, we stress that there do not exist quantized stable eigenvalues near the origin, which is very different from the spectrum of linearized operator in critical/supercritical blowup regime [21, 11, 9, 10, 35]. However, we expect quantized eigenval… view at source ↗

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Works this paper leans on

51 extracted references · 44 canonical work pages

[1]

Self-similar blow-up profiles for slightly supercritical nonlinear Schrödinger equations.Ann

Yakine Bahri, Yvan Martel, and Pierre Raphaël. Self-similar blow-up profiles for slightly supercritical nonlinear Schrödinger equations.Ann. Henri Poincaré, 22(5):1701–1749, 2021

2021
[2]

Turán type inequalities for modified Bessel functions.Bull

Árpád Baricz. Turán type inequalities for modified Bessel functions.Bull. Aust. Math. Soc., 82(2):254–264, 2010

work page 2010
[3]

New estimates for a time-dependent schrödinger equation.Duke Mathemat- ical Journal, 159(3):417–477, 2011

Marius Beceanu. New estimates for a time-dependent schrödinger equation.Duke Mathemat- ical Journal, 159(3):417–477, 2011

work page 2011
[4]

Universi- text

Haim Brezis.Functional analysis, Sobolev spaces and partial differential equations. Universi- text. New York, NY: Springer, 2011

work page 2011
[5]

V. S. Buslaev and G. S. Perelman. On the stability of solitary waves for nonlinear Schrödinger equations. InNonlinear evolution equations, volume 164 ofAmer. Math. Soc. Transl. Ser. 2, pages 75–98. Amer. Math. Soc., Providence, RI, 1995

work page 1995
[6]

New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003

Thierry Cazenave.Semilinear Schrödinger equations, volume 10 ofCourant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003

work page 2003
[7]

Spectra of lin- earized operators for nls solitary waves.SIAM Journal on Mathematical Analysis, 39(4):1070– 1111, 2008

Shu-Ming Chang, Stephen Gustafson, Kenji Nakanishi, and Tai-Peng Tsai. Spectra of lin- earized operators for nls solitary waves.SIAM Journal on Mathematical Analysis, 39(4):1070– 1111, 2008

work page 2008
[8]

Co-dimension one stable blowup for the quadratic wave equation beyond the light cone.Com- mun

Po-Ning Chen, Roland Donninger, Irfan Glogić, Michael McNulty, and Birgit Schörkhuber. Co-dimension one stable blowup for the quadratic wave equation beyond the light cone.Com- mun. Math. Phys., 405(2):46, 2024. Id/No 34

work page 2024

Show all 51 references

[9]

Spectral analysis for singularity formation of the two dimensional Keller-Segel system.Ann

Charles Collot, Tej-Eddine Ghoul, Nader Masmoudi, and Van Tien Nguyen. Spectral analysis for singularity formation of the two dimensional Keller-Segel system.Ann. PDE, 8(1):Paper No. 5, 74, 2022

2022
[10]

Singularity formed by the collision of two collapsing solitons in interaction for the 2d Keller-Segel system

Charles Collot, Tej-Eddine Ghoul, Nader Masmoudi, and Van Tien Nguyen. Singularity formed by the collision of two collapsing solitons in interaction for the 2d Keller-Segel system. Preprint, arXiv:2409.05363 [math.AP] (2024), 2024

2024 arXiv
[11]

Strongly anisotropic type II blow up at an isolated point.J

Charles Collot, Frank Merle, and Pierre Raphaël. Strongly anisotropic type II blow up at an isolated point.J. Amer. Math. Soc., 33(2):527–607, 2020. 154 Z. LI

2020
[12]

On the stability of type I blow up for the energy super critical heat equation.Mem

Charles Collot, Pierre Raphaël, and Jeremie Szeftel. On the stability of type I blow up for the energy super critical heat equation.Mem. Amer. Math. Soc., 260(1255):v+97, 2019

2019
[13]

Mode stability of self-similar wave maps in higher dimensions.Comm

Ovidiu Costin, Roland Donninger, and Irfan Glogić. Mode stability of self-similar wave maps in higher dimensions.Comm. Math. Phys., 351(3):959–972, 2017

2017
[14]

OvidiuCostin, RolandDonninger, IrfanGlogić, andMinHuang.Onthestabilityofself-similar solutions to nonlinear wave equations.Comm. Math. Phys., 343(1):299–310, 2016

2016
[15]

RolandDonninger.Spectraltheoryandself-similarblowupinwaveequations. Bull. Am. Math. Soc., New Ser., 61(4):659–685, 2024

2024
[16]

On blowup in supercritical wave equations.Comm

Roland Donninger and Birgit Schörkhuber. On blowup in supercritical wave equations.Comm. Math. Phys., 346(3):907–943, 2016

2016
[17]

Fedoryuk.Asymptotic analysis: linear ordinary differential equations

Mikhail V. Fedoryuk.Asymptotic analysis: linear ordinary differential equations. Translated from the Russian by Andrew Rodick. Berlin: Springer-Verlag, 1993

1993
[18]

Singular ring solutions of critical and super- critical nonlinear Schrödinger equations.Phys

Gadi Fibich, Nir Gavish, and Xiao-Ping Wang. Singular ring solutions of critical and super- critical nonlinear Schrödinger equations.Phys. D, 231(1):55–86, 2007

2007
[19]

Nonlinear stability of homothetically shrinking Yang- Mills solitons in the equivariant case.Comm

Irfan Glogić and Birgit Schörkhuber. Nonlinear stability of homothetically shrinking Yang- Mills solitons in the equivariant case.Comm. Partial Differential Equations, 45(8):887–912, 2020

2020
[20]

Stable Singularity Formation for the Keller–Segel System in Three Dimensions.Arch

Irfan Glogić and Birgit Schörkhuber. Stable Singularity Formation for the Keller–Segel System in Three Dimensions.Arch. Ration. Mech. Anal., 248(1):4, 2024

2024
[21]

On melting and freezing for the 2D radial Stefan problem

Mahir Hadžić and Pierre Raphaël. On melting and freezing for the 2D radial Stefan problem. J. Eur. Math. Soc. (JEMS), 21(11):3259–3341, 2019

2019
[22]

A solution to the focusing 3d NLS that blows up on a contracting sphere.Trans

Justin Holmer, Galina Perelman, and Svetlana Roudenko. A solution to the focusing 3d NLS that blows up on a contracting sphere.Trans. Amer. Math. Soc., 367(6):3847–3872, 2015

2015
[23]

Mourad E. H. Ismail. Bessel functions and the infinite divisibility of the Studentt-distribution. Ann. Probability, 5(4):582–585, 1977

1977
[24]

Sharp universal rate for stable blow-up of corotational wave maps.Comm

Kihyun Kim. Sharp universal rate for stable blow-up of corotational wave maps.Comm. Math. Phys., 402(3):2387–2463, 2023

2023
[25]

Self-similar solutions to super-critical gKdV.Nonlinearity, 28(3):545–575, 2015

Herbert Koch. Self-similar solutions to super-critical gKdV.Nonlinearity, 28(3):545–575, 2015

2015
[26]

Krieger and W

J. Krieger and W. Schlag. Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension.J. Amer. Math. Soc., 19(4):815–920, 2006

2006
[27]

Uniqueness of positive solutions of∆u − u + up = 0 in RN

Man Kam Kwong. Uniqueness of positive solutions of∆u − u + up = 0 in RN. Archive for Rational Mechanics and Analysis, 105(3):243–266, 1989

1989
[28]

Stability of isotropic singularities for the nonlinear schrödinger equation.Physica D: Nonlinear Phenomena, 47(3):393–415, 1991

MJ Landman, GC Papanicolaou, Catherine Sulem, Pierre Louis Sulem, and Xiao Ping Wang. Stability of isotropic singularities for the nonlinear schrödinger equation.Physica D: Nonlinear Phenomena, 47(3):393–415, 1991

1991
[29]

Focusing and multi-focusing solu- tions of the nonlinear schrödinger equation.Physica D: Nonlinear Phenomena, 31(1):78–102, 1988

BJ LeMesurier, G Papanicolaou, C Sulem, and PL Sulem. Focusing and multi-focusing solu- tions of the nonlinear schrödinger equation.Physica D: Nonlinear Phenomena, 31(1):78–102, 1988

1988
[30]

Mode stability for self-similar blowup of slightly supercritical nls: II

Zexing Li. Mode stability for self-similar blowup of slightly supercritical nls: II. high-energy spectrum. arxiv preprint 2025

2025
[31]

On stability of self-similar blowup for mass supercritical nls

Zexing Li. On stability of self-similar blowup for mass supercritical nls. arXiv preprint arXiv:2304.02078v2, 2023

2023 arXiv
[32]

Nonradial stability of self-similar blowup to Keller-Segel equation in three dimensions

Zexing Li and Tao Zhou. Nonradial stability of self-similar blowup to Keller-Segel equation in three dimensions. Preprint, arXiv:2501.07073 [math.AP] (2025), 2025

2025 arXiv
[33]

Focusing singularity of the cubic schrödinger equation.Physical Review A, 34(2):1200, 1986

David W McLaughlin, George C Papanicolaou, Catherine Sulem, and Pierre-Louis Sulem. Focusing singularity of the cubic schrödinger equation.Physical Review A, 34(2):1200, 1986

1986
[34]

Blow up of the critical norm for some radialL2 super critical nonlinear Schrödinger equations.Amer

Frank Merle and Pierre Raphaël. Blow up of the critical norm for some radialL2 super critical nonlinear Schrödinger equations.Amer. J. Math., 130(4):945–978, 2008

2008
[35]

FrankMerle, PierreRaphaël, andIgorRodnianski.TypeIIblowupfortheenergysupercritical NLS. Camb. J. Math., 3(4):439–617, 2015

2015
[36]

Stable self-similar blow-up dynamics for slightly L2 super-critical NLS equations.Geom

Frank Merle, Pierre Raphaël, and Jeremie Szeftel. Stable self-similar blow-up dynamics for slightly L2 super-critical NLS equations.Geom. Funct. Anal., 20(4):1028–1071, 2010

2010
[37]

On collapsing ring blow-up solutions to the mass supercritical nonlinear Schrödinger equation.Duke Math

Frank Merle, Pierre Raphaël, and Jeremie Szeftel. On collapsing ring blow-up solutions to the mass supercritical nonlinear Schrödinger equation.Duke Math. J., 163(2):369–431, 2014

2014
[38]

F. W. J. Olver. Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders.J. Res. Nat. Bur. Standards Sect. B, 63B:131–169, 1959

1959
[39]

Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark, editors. NIST handbook of mathematical functions. U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge,
[40]

On the formation of singularities in solutions of the critical nonlinear Schrödinger equation.Ann

Galina Perelman. On the formation of singularities in solutions of the critical nonlinear Schrödinger equation.Ann. Henri Poincaré, 2(4):605–673, 2001

2001
[41]

Existence and stability of a solution blowing up on a sphere for an L2- supercritical nonlinear Schrödinger equation.Duke Math

Pierre Raphaël. Existence and stability of a solution blowing up on a sphere for an L2- supercritical nonlinear Schrödinger equation.Duke Math. J., 134(2):199–258, 2006

2006
[42]

Michael Reed and Barry Simon.Methods of modern mathematical physics. III. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. Scattering theory

1979
[43]

Elsevier, 1972

Michael Reed, Barry Simon, Barry Simon, and Barry Simon.Methods of modern mathematical physics, volume 4. Elsevier, 1972

1972
[44]

Stein and Guido Weiss.Introduction to Fourier analysis on Euclidean spaces, volume No

Elias M. Stein and Guido Weiss.Introduction to Fourier analysis on Euclidean spaces, volume No. 32 ofPrinceton Mathematical Series. Princeton University Press, Princeton, NJ, 1971

1971
[45]

Focusing nonlinear Schrödinger equation and wave- packet collapse

Catherine Sulem and Pierre-Louis Sulem. Focusing nonlinear Schrödinger equation and wave- packet collapse. InProceedings of the Second World Congress of Nonlinear Analysts, Part 2 (Athens, 1996), volume 30, pages 833–844, 1997

1996
[46]

Springer-Verlag, New York, 1999

Catherine Sulem and Pierre-Louis Sulem.The nonlinear Schrödinger equation, volume 139 of Applied Mathematical Sciences. Springer-Verlag, New York, 1999. Self-focusing and wave collapse

1999
[47]

Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006

Terence Tao.Nonlinear dispersive equations, volume 106 ofCBMS Regional Conference Series in Mathematics. Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis

2006
[48]

G. N. Watson.A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge; The Macmillan Company, New York, 1944

1944
[49]

Weinstein

Michael I. Weinstein. Modulational stability of ground states of nonlinear Schrödinger equa- tions. SIAM J. Math. Anal., 16(3):472–491, 1985

1985
[50]

Mode stability of blow-up for wave maps in the absence of symmetry

Max Weissenbacher, Herbert Koch, and Roland Donninger. Mode stability of blow-up for wave maps in the absence of symmetry. Preprint, arXiv:2503.02632 [math.AP] (2025), 2025. Laboratoire Analyse, Géométrie et Modélisation, CY Cergy Paris Université, 2 a venue Adolphe Chauvin, 9...

2025 arXiv
[2010]

With 1 CD-ROM (Windows, Macintosh and UNIX). 155

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Self-similar blow-up profiles for slightly supercritical nonlinear Schrödinger equations.Ann

Yakine Bahri, Yvan Martel, and Pierre Raphaël. Self-similar blow-up profiles for slightly supercritical nonlinear Schrödinger equations.Ann. Henri Poincaré, 22(5):1701–1749, 2021

2021

[2] [2]

Turán type inequalities for modified Bessel functions.Bull

Árpád Baricz. Turán type inequalities for modified Bessel functions.Bull. Aust. Math. Soc., 82(2):254–264, 2010

work page 2010

[3] [3]

New estimates for a time-dependent schrödinger equation.Duke Mathemat- ical Journal, 159(3):417–477, 2011

Marius Beceanu. New estimates for a time-dependent schrödinger equation.Duke Mathemat- ical Journal, 159(3):417–477, 2011

work page 2011

[4] [4]

Universi- text

Haim Brezis.Functional analysis, Sobolev spaces and partial differential equations. Universi- text. New York, NY: Springer, 2011

work page 2011

[5] [5]

V. S. Buslaev and G. S. Perelman. On the stability of solitary waves for nonlinear Schrödinger equations. InNonlinear evolution equations, volume 164 ofAmer. Math. Soc. Transl. Ser. 2, pages 75–98. Amer. Math. Soc., Providence, RI, 1995

work page 1995

[6] [6]

New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003

Thierry Cazenave.Semilinear Schrödinger equations, volume 10 ofCourant Lecture Notes in Mathematics. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003

work page 2003

[7] [7]

Spectra of lin- earized operators for nls solitary waves.SIAM Journal on Mathematical Analysis, 39(4):1070– 1111, 2008

Shu-Ming Chang, Stephen Gustafson, Kenji Nakanishi, and Tai-Peng Tsai. Spectra of lin- earized operators for nls solitary waves.SIAM Journal on Mathematical Analysis, 39(4):1070– 1111, 2008

work page 2008

[8] [8]

Co-dimension one stable blowup for the quadratic wave equation beyond the light cone.Com- mun

Po-Ning Chen, Roland Donninger, Irfan Glogić, Michael McNulty, and Birgit Schörkhuber. Co-dimension one stable blowup for the quadratic wave equation beyond the light cone.Com- mun. Math. Phys., 405(2):46, 2024. Id/No 34

work page 2024