Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curves

Niek Lamoree; Oliver Fabert

REVIEW 1 major objections 4 minor 32 references

Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curves

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that Hamiltonian PDEs with regularizing nonlinearities and Diophantine-admissible periods admit forced time-periodic solutions, obtained as limits of finite-dimensional pseudoholomorphic curve solutions.

desk verdict A serious, ambitious attempt at infinite-dimensional Floer compactness for Hamiltonian PDEs, but the key bubbling-off estimate in Lemma 6.2 has a real gap; worth a referee's time, but the main theorem is not yet proven. read the letter →

arxiv 1908.03165 v4 pith:7HP2K7TC submitted 2019-08-08 math.SG math.AP

Oliver Fabert , Niek Lamoree This is my paper · ORCID

classification math.SGmath.AP MSC 53D4037K5535B10

keywords HamiltonianPDEstime-periodicsolutionspseudoholomorphiccurvessmalldivisorproblemDiophantineapproximationregularizingnonlinearitiesHilbertscalecup-lengthestimate

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 tries to bring the pseudoholomorphic-curve method of finite-dimensional Hamiltonian dynamics into the infinite-dimensional world of Hamiltonian PDEs. It claims that when the linear part has eigenvalues $a n^d$ and the nonlinearity is sufficiently regularizing, meaning its high-frequency Fourier coefficients decay at order $n^{-h}$ with $h>dr$, where $r$ measures how well the period ratio $aT/2\pi$ can be approximated by rationals, the associated Cauchy-Riemann-type gradient equation has a solution in the full Hilbert space, connecting the trivial zero solution to a weak forced $T$-periodic solution of the original PDE. If correct, this yields a single existence mechanism for forced periodic solutions of regularized nonlinear wave and nonlinear Schrödinger equations, with an explicit smoothness count, and a cup-length estimate giving several distinct solutions when the phase space is a product of a closed symplectic manifold with a linear Hilbert space. The wider interest is that a compactness theorem of this kind is what a homology theory counting periodic solutions of PDEs would be built on.

What carries the argument

The engine is a comparison between two rates of decay. For an admissible operator with eigenvalues $a n^d$ and admissible periods, the small-divisor estimate $\inf_{p\in\mathbb Z}\left|\frac{2\pi p}{T}-a n^d\right|\ge c\,n^{-d(r-1)}$ says the denominators in the Fourier-mode ODE are never smaller than a fixed power of $n$, where $r$ is the irrationality measure of $aT/2\pi$. The $A$-admissibility condition on the nonlinearity forces its gradient's Fourier coefficients to be $o(n^{-h})$ with $h>dr$, so the high modes of the curve are killed before the small denominators can amplify them. Splitting a truncated solution into low and high frequencies, the high tail is shown to vanish uniformly as the truncation grows, while a bubbling-off argument plus elliptic bootstrapping keeps all derivatives bounded; the two together produce the compactness theorem.

What would settle it

Run the Fourier-mode computation of Example 2.5 on the forced linear wave equation: set the forcing Fourier coefficients equal to the vanishing denominator $(T/X-p/n)(T/X+p/n)$ at the best rational approximants $p/n$ of $T/X$; for a non-Diophantine ratio the forcing is smooth but the solution coefficients do not decay, so no $T$-periodic solution exists. If the same obstruction appeared for a Diophantine ratio with $h>dr$, the main theorem would be false; a reader can check this directly by computing the solution coefficients for a fixed Diophantine $T/X$ and comparing their decay with $h-d(r-1)$.

Watch

Extended reading notes

Core claim

Main Theorem 4.1 states: for a Hamiltonian PDE with $A$-admissible nonlinearity $G_t$, there exists a $(\lfloor h/d\rfloor-1)$-times differentiable map $\tilde u:\mathbb R\times\mathbb R\to H_{h-d(r-1)-1/2}$ satisfying the equation $\bar\partial \tilde u+\varphi(s)\nabla G_t(\tilde u)=0$ and the periodicity $\tilde u(s,t+T)=\varphi^A_{-T}\tilde u(s,t)$, with limits $0$ at $s\to-\infty$ and a weak $T$-periodic solution $u_1$ at $s\to+\infty$. The proof is carried by Theorem 8.1, an infinite-dimensional compactness statement: finitely truncated solutions of the same equation $C^{\lfloor h/d\rfloor-1}_{\mathrm{loc}}$-converge, after passing to a subsequence, to such a solution in the full Hilbert space. The number $h-d(r-1)-\tfrac12$ is the paper's explicit answer to how much regularity the nonlinearity must possess relative to the Diophantine quality of the periods; when the nonlinearity is $\infty$-regularizing, the curve and the periodic solution are smooth in all variables.

Load-bearing premise

The load-bearing assumption is that the ratio of time period to space period cannot be approximated too well by rationals, and that the nonlinearity smooths high frequencies fast enough relative to that approximation rate; if the ratio is too well approximable, the paper's own counterexample shows a smooth forcing term with no periodic solution.

Editorial extensions

If this is right

Any weakly $A$-admissible nonlinearity, meaning a time-periodic $h$-regularizing forcing with $h>dr$ up to a bounded cut-off, still forces a $T$-periodic solution of the PDE, of regularity $h-d(r-1)-\tfrac12$.
For generic time periods, where the irrationality measure is $2$ and Diophantine numbers have full measure, the condition reduces to $h>2d$; the paper's examples give a strong solution for the regularized nonlinear wave equation when $h>3.5$ and for the regularized nonlinear Schrödinger equation when $h>5$.
When the nonlinearity is $\infty$-regularizing, the periodic solution is smooth in time and space, so the existence theorem covers nonlocal sine-Gordon and nonlocal Schrödinger models with smooth kernels without an integrability assumption.
On a phase space $M\times H$ with $M$ closed and $\pi_2(M)=0$, the method yields at least $\mathrm{cl}(M)$ distinct $T$-periodic solutions, ordered by symplectic action.

Reading between the lines

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

Editorial inference: because inequality (13) is the only place number theory enters, the same compactness machinery should transfer to any PDE whose linear frequencies admit a comparable Diophantine lower bound, such as higher-dimensional tori with a multi-index approximation condition.
Editorial inference: the explicit regularity formula $h-d(r-1)-\tfrac12$ makes a testable prediction: for a fixed smoothing kernel of order $h$, increasing the irrationality measure of $T/X$ should visibly lower the differentiability class of the forced periodic response; a numerical Fourier-mode study could verify this.
Editorial inference: the dimension-free convergence statement is the ingredient from which genuine Morse-theoretic counts of PDE periodic orbits could be built, going beyond the single cup-length example treated here; the paper states such a homology theory as an ongoing project but does not construct it.

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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

Dear colleague,

You should know the main thing about this paper: it is a serious, substantial attempt to extend Floer theory to infinite-dimensional Hamiltonian PDEs, and it contains a genuinely new compactness framework. But the central proof as written has a gap in Lemma 6.2, the bubbling-off estimate that supplies the uniform C^1 bound. That gap is load-bearing, so the main theorem is not fully proven yet.

What is genuinely new: Theorem 8.1, an infinite-dimensional Gromov-Floer compactness theorem for linear symplectic Hilbert spaces, using Diophantine control on the small divisors to force the high-frequency tails to vanish. The cup-length estimate for M×H in Section 10 is also new. The authors write honestly, cite the relevant finite-dimensional Floer theory, and do not hide limitations: they state when the solution is only weak, include a counterexample showing some Diophantine condition is necessary, and take care to distinguish A-admissible from weakly A-admissible nonlinearities.

The main problem is in Section 6. In the proof of Lemma 6.2, after rescaling by C_k, they claim

∫_{√C/2}^{√C} r L(γ_r)^2 dr ≤ 10πT ||F||_{C0}.

The step from the area bound to this inequality is not justified. The standard length-area estimate in this normalization carries L^2/r dr, not r L^2 dr. With the r-weight, the left-hand side is a weighted second moment of the derivative, which is not controlled by area or energy for a disk whose radius goes to infinity. A model linear map with |∂s v(0)|=1 makes the left side of order C^2, so the claimed uniform bound cannot hold without an additional argument. This gap breaks the proof that L(γ_r)→0, hence that the area of the shrinking disks goes to 0, and hence the C^1 bound and the compactness theorem lose their support. This looks repairable by a correct local-energy or monotonicity argument, but it is a real hole.

A smaller issue: Corollary 9.5 states h>3 1/2 and then says "h>3 suffices." That is internally inconsistent; presumably "h>3.5" is meant. The cup-length section is also sketchier, with several lemmas deferred to "analogous" reasoning.

Net: the paper deserves a serious referee, not a desk rejection. The ideas are significant and the framework may well be right, but the proof needs major repair before the main theorem can be considered verified.

Referee Report

1 major / 4 minor

Summary. The paper proposes an infinite-dimensional extension of Gromov-Floer compactness for Floer curves associated with Hamiltonian PDEs of the form ˙u = JAu + J∇Ft(u), where A is a self-adjoint operator with eigenvalues a n^d and the nonlinearity is h-regularizing with h > dr, r being the irrationality measure of aT/2π. The main result, Theorem 4.1, asserts the existence of a Floer curve satisfying ∂u + φ(s)∇G_t(u)=0 together with the φ^A_T-periodicity condition, and connecting the trivial solution to a weak T-periodic solution of the PDE. The proof proceeds by finite-dimensional approximation, a bubbling-off analysis to bound derivatives, a Diophantine small-divisor estimate to control high-frequency tails, and a diagonal compactness argument. The paper also extends the existence result to weakly A-admissible nonlinearities and proves a cup-length estimate for the number of periodic solutions when the phase space is M × H with M a closed symplectic manifold with π2(M)=0.

Significance. If the technical proof can be completed, the paper would be a substantial contribution: it would give a genuine infinite-dimensional analogue of Gromov-Floer compactness, connect symplectic topology to forced Hamiltonian PDEs, provide explicit regularity thresholds in terms of Diophantine properties, and yield new periodic-solution results for nonlocal wave and Schrödinger equations. The paper is also commendable for stating precise hypotheses, giving a counterexample (Example 2.5) showing the need for the Diophantine condition, and being explicit that the finite-dimensional Floer machinery is cited rather than redeveloped. The central claim, however, rests on a key scaling estimate in Lemma 6.2 that is not justified and appears to be incorrect as written; until that estimate is repaired, the compactness theorem and the main existence theorem are not established.

major comments (1)

[§6, Lemma 6.2] The length-area estimate in the proof of Lemma 6.2 has a scaling problem that is load-bearing. After the rescaling v_k(z)=u_k(z/C_k+z_k), the proof claims ∫_{√C_k/2}^{√C_k} r L(γ_r^k)^2 dr ≤ 10πT||F||_{C^0}. This inequality is not a consequence of the preceding bounds and is dimensionally inconsistent. For the model map v_k(z)=z on the rescaled disk, one has |∂s v_k(0)|=1 and |∂θ v_k|=r, so L(γ_r^k)=2πr and the left-hand side is of order C_k^2, not O(1). The standard length-area estimate in this normalization carries r^{-1}, not r, and with the r-weight the claimed uniform bound cannot hold. Since this estimate is the only mechanism in the proof forcing L_0^k→0, the conclusion L_0^k→0 and hence the uniform C^1 bound on the Floer curves are unsupported. Proposition 6.3, Proposition 7.2, and Theorem 8.1 all inherit this gap. The argument may be repairable by a different local-energy or monotonicity estimate, but as written the central compactness proof is incomplete.

minor comments (4)

[Introduction, p. 2] There is a typo: “inﬁninte-dimensional” should read “infinite-dimensional.”
[§8, proof of Proposition 8.4] The sentence “Observe that the regularity requirements stated above ensure that the singles-derivative also exists” contains a typo (“singles-derivative” should be “s-derivative”).
[§2, Examples 2.2 and 2.3] The displayed eigenvalues are written as “λn = and” and “λn =and”; the intended expression appears to be λ_n = a n^d, and the formulas should be corrected.
[§7, equation (14)] The uniformity in k of the limit (14) is asserted rather than proved; after Lemma 6.2 is repaired, the proof should explicitly justify that the use of the C^m bounds from Proposition 6.3 is uniform in k.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; conditional compactness proof, self-citations only contextual.

full rationale

The claimed derivation is not circular. Main Theorem 4.1 is obtained by: (i) producing finite-dimensional Floer curves u_k via standard finite-dimensional Floer theory (Proposition 5.2, citing [Sal97], [DS94], [MS04]); (ii) proving uniform C^m bounds by a bubbling-off contradiction and elliptic bootstrapping (Lemma 6.2 / Proposition 6.3); (iii) estimating the Fourier tail of u_k using the Diophantine lower bound (13), Lemma 7.1, and the h-regularizing decay (14) (Proposition 7.2); and (iv) taking a diagonal subsequence (Theorem 8.1) and extracting asymptotic orbits from finite energy (Theorem 8.2). Each step uses hypotheses stated before the conclusion: h>dr, admissibility of (T,X), bounded support, and h-regularizing behaviour. No parameter is fitted to the target solution, and the target periodic orbit is not assumed in any definition. The Diophantine condition is explicitly a condition on the period data and is shown necessary by Example 2.5; it is not derived from the existence claim. The regularity of the output, h-d(r-1)-1/2, is weaker than the input regularity h and is obtained from the coefficient decay estimate (15), not by renaming the input. Self-citations are present but not load-bearing: [Fab21] is used only for comparison and for the NLSE example, and [FL21] only to contextualize the particle-field class; the compactness statement is proved in Sections 6-8 rather than imported. Finite-dimensional Floer theory is an external body of results, not the authors' own prior work. A possible scaling defect in Lemma 6.2, if real, would be a correctness gap in the proof of the C^1 bound, not a circular identification: no equation in that argument is equivalent to the theorem by construction. Overall the paper is a conditional existence proof whose high-frequency control is supplied by an explicit small-divisor estimate.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The proof rests on the stated structural assumptions: admissible A, Diophantine periods, h-regularizing nonlinearities with h>dr, and bounded support. It also relies on standard finite-dimensional Floer theory and Diophantine approximation. No data are fitted and no new physical entities are postulated.

assumptions (7)

domain assumption The linear operator A is admissible: pure of degree d with eigenvalues λ_n = a n^d and [J,A]=0 (Definition 2.1).
Used throughout to diagonalize the free flow and reduce small divisors to Diophantine properties of aT/(2π). It holds for the NLW and NLS examples but is not general.
domain assumption The periods (T,X) are admissible, meaning aT/(2π) is Diophantine with finite irrationality measure r (Definition 2.4).
Provides the lower bound in inequality (13) and ensures the free flow has only the trivial fixed point; the counterexample in Example 2.5 shows failure without it.
domain assumption The nonlinearity is h-regularizing with h>dr and bounded C^α norms on H_{-h} (Definition 3.3 conditions 2 and 3).
Ensures ∇F maps H into H_h with Fourier decay, used in Lemma 6.1 and Proposition 7.2 to make the high-frequency tails vanish.
domain assumption A-admissible nonlinearities satisfy the bounded support condition (Definition 3.3 condition 4).
Needed for the maximum principle in Proposition 5.1 to trap Floer curves in a compact set; later removed for weakly admissible nonlinearities via a cutoff argument.
standard math Finite-dimensional Floer theory is taken as established, including existence of Floer curves, Gromov-Floer compactness, and transversality results (Section 5, Proposition 5.2, citing [Sal97], [DS94], [MS04], [FH94], [Oan04], [Wen10]).
The infinite-dimensional proof reduces to finite-dimensional Floer curves and uses their existence as a black box.
standard math Diophantine approximation and irrationality measure properties (Section 2, citing [Bug12], [Sal08]).
Gives the full-measure property of Diophantine numbers and the approximation lower bound used in inequality (13).
standard math Hilbert scale, Sobolev embedding, and sc-smooth analysis background (Section 3 and Appendix A, citing [Kuk00], [HWZ10], [Bre10]).
Used for regularity and bootstrap arguments, including the dimension-independent Sobolev embedding used in Proposition 6.3.

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Pith. "Pith review of Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curves." pith.science (2026). https://pith.science/paper/7HP2K7TC

@misc{pith2026190803165,
  author       = {Pith},
  title        = {Pith review of: Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7HP2K7TC}},
  note         = {Machine review of arXiv:1908.03165}
}

read the original abstract

We extend the pseudoholomorphic curve methods from Floer theory to infinite-dimensional phase spaces and use our results to prove the existence of a forced time-periodic solution to a general Hamiltonian PDE with regularizing nonlinearity. In particular, when the nonlinearity is sufficiently regularizing, bounded and time-periodic, we prove an infinite-dimensional version of Gromov-Floer compactness by using ideas from the theory of Diophantine approximations to overcome the small divisor problem. Furthermore, in the case when the infinite-dimensional phase space is a product of a finite-dimensional closed symplectic manifold with linear symplectic Hilbert space, we prove a cup-length estimate for the number of periodic solutions.

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Works this paper leans on

32 extracted references · 32 canonical work pages

[1]

Free vibrations for a nonlinear wave equation and a theorem of P

Haïm Br \' e zis, Jean-Michel Coron, and Louis Nirenberg. Free vibrations for a nonlinear wave equation and a theorem of P . R abinowitz. Communications on Pure and Applied Mathematics , 33(5):667--684, September 1980

work page 1980
[2]

M. Berti. Nonlinear Oscillations of Hamiltonian PDEs . Birkh\" a user Boston, 2007

work page 2007
[3]

H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations . Springer New York, 2010

work page 2010
[4]

Distribution Modulo One and Diophantine Approximation

Yann Bugeaud. Distribution Modulo One and Diophantine Approximation . Cambridge Tracts in Mathematics. Cambridge University Press, 2012

work page 2012
[5]

Eugene Wayne

Walter Craig and C. Eugene Wayne. Newton's method and periodic solutions of nonlinear wave equations. Communications on Pure and Applied Mathematics , 46(11):1409--1498, December 1993

work page 1993
[6]

Di Gregorio

L. Di Gregorio. Infinite dimensional Hamiltonian systems and nonlinear wave equation: periodic orbits with long minimal period . PhD thesis, Roma Tre University, 2005

work page 2005
[7]

Dostoglou and D

S. Dostoglou and D. Salamon. Self-Dual Instantons and Holomorphic Curves . Annals of Mathematics , 139(3):581--640, May 1994

work page 1994
[8]

Hakan Eliasson, Benoît Grébert, and Sergeï B

L. Hakan Eliasson, Benoît Grébert, and Sergeï B. Kuksin. KAM for the non-linear B eam equation 2: A normal form theorem , 2015

work page 2015

Show all 32 references

[9]

Hakan Eliasson, Benoît Grébert, and Sergeï B

L. Hakan Eliasson, Benoît Grébert, and Sergeï B. Kuksin. A KAM theorem for space-multidimensional Hamiltonian PDEs . Proc. Steklov Inst. Math. , 295(1):129--147, Nov 2016

2016
[10]

Cemal Eringen, editor

A. Cemal Eringen, editor. Nonlocal Continuum Field Theories . Springer New York, 2004

2004
[11]

O. Fabert. Hamiltonian F loer theory for nonlinear S chrödinger equations and the small divisor problem. International Mathematical Research Notices , 2021

2021
[12]

Floer and H

A. Floer and H. Hofer. Symplectic homology I . O pen S ets in C n. Mathematische Zeitschrift , 215(1):37--88, 1994

1994
[13]

Cuplength estimates for periodic solutions of Hamiltonian particle-field systems

Oliver Fabert and Niek Lamoree. Cuplength estimates for periodic solutions of Hamiltonian particle-field systems . ArXiv , July 2021

2021
[14]

Symplectic fixed points and holomorphic spheres

Andreas Floer. Symplectic fixed points and holomorphic spheres . Commun. Math. Phys. , 120(4):575--611, Dec 1989

1989
[15]

Periodic solutions for the S chr\" o dinger equation with nonlocal smoothing nonlinearities in higher dimension

Guido Gentile and Michela Procesi. Periodic solutions for the S chr\" o dinger equation with nonlocal smoothing nonlinearities in higher dimension. Journal of Differential Equations , 245(11):3253--3326, December 2008

2008
[16]

Periodic S olutions for a C lass of N onlinear P artial D ifferential E quations in H igher D imension

Guido Gentile and Michela Procesi. Periodic S olutions for a C lass of N onlinear P artial D ifferential E quations in H igher D imension. Communications in Mathematical Physics , 289(3):863--906, Aug 2009

2009
[17]

M. Gromov. Pseudo holomorphic curves in symplectic manifolds . Invent. Math. , 82(2):307--347, Jun 1985

1985
[18]

Time-dependent variational approach for Bose-Einstein condensates with nonlocal interaction

Fernando Haas and Bengt Eliasson. Time-dependent variational approach for Bose-Einstein condensates with nonlocal interaction . J. Phys. B: At. Mol. Opt. Phys. , 51(17):175302, Aug 2018

2018
[19]

Hofer, K

H. Hofer, K. Wysocki, and E. Zehnder. Sc-smoothness, retractions and new models for smooth spaces. Discrete and Continuous Dynamical Systems , 28(2):665--788, April 2010

2010
[20]

S. B. Kuksin. Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum. Functional Analysis and Its Applications , 21(3):192--205, Jul 1987

1987
[21]

S. B. Kuksin. Infinite-dimensional symplectic capacities and a squeezing theorem for H amiltonian P D E 's. Communications in Mathematical Physics , 167(3):531--552, Feb 1995

1995
[22]

S. B. Kuksin. Analysis of Hamiltonian PDEs . Oxford University Press, Sep 2000

2000
[23]

McDuff and D

D. McDuff and D. Salamon. J-holomorphic Curves and Symplectic Topology . Number v. 52 in American Mathematical Society colloquium publications. American Mathematical Society, 2004

2004
[24]

A. Oancea. A survey of Floer homology for manifolds with contact type boundary or symplectic homology . Ensaios Matem \'a ticos , 7:51--91, September 2004

2004
[25]

P. H. Rabinowitz. Free vibrations for a semilinear wave equation. Communications on Pure and Applied Mathematics , 31(1):31--68, 1978

1978
[26]

D. Salamon. Lectures on Floer homology , 1997. URL: https://people.math.ethz.ch/ salamon/PREPRINTS/floer.pdf

1997
[27]

V. Kh. Salikhov. On the irrationality measure of pi . Russian Mathematical Surveys , 63(3):570--572, jun 2008

2008
[28]

A quantum cup-length estimate for symplectic fixed points

Matthias Schwarz. A quantum cup-length estimate for symplectic fixed points. Inventiones Mathematicae , 133(2):353--397, July 1998

1998
[29]

E. T. Tomboulis. Nonlocal and quasilocal field theories . Phys. Rev. D , 92(12):125037, Dec 2015

2015
[30]

Eugene Wayne

C. Eugene Wayne. Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Comm. Math. Phys. , 127(3):479--528, 1990

1990
[31]

C. Wendl. A beginner's overview of symplectic homology , 2010. URL: https://www.mathematik.hu-berlin.de/ wendl/pub/SH.pdf

2010
[32]

Chaoticons described by nonlocal nonlinear Schrödinger equation

Lanhua Zhong, Yuqi Li, Yong Chen, Weiyi Hong, Wei Hu, and Qi Guo. Chaoticons described by nonlocal nonlinear Schrödinger equation . Sci. Rep. , 7:41438, Jan 2017

2017

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Free vibrations for a nonlinear wave equation and a theorem of P

Haïm Br \' e zis, Jean-Michel Coron, and Louis Nirenberg. Free vibrations for a nonlinear wave equation and a theorem of P . R abinowitz. Communications on Pure and Applied Mathematics , 33(5):667--684, September 1980

work page 1980

[2] [2]

M. Berti. Nonlinear Oscillations of Hamiltonian PDEs . Birkh\" a user Boston, 2007

work page 2007

[3] [3]

H. Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations . Springer New York, 2010

work page 2010

[4] [4]

Distribution Modulo One and Diophantine Approximation

Yann Bugeaud. Distribution Modulo One and Diophantine Approximation . Cambridge Tracts in Mathematics. Cambridge University Press, 2012

work page 2012

[5] [5]

Eugene Wayne

Walter Craig and C. Eugene Wayne. Newton's method and periodic solutions of nonlinear wave equations. Communications on Pure and Applied Mathematics , 46(11):1409--1498, December 1993

work page 1993

[6] [6]

Di Gregorio

L. Di Gregorio. Infinite dimensional Hamiltonian systems and nonlinear wave equation: periodic orbits with long minimal period . PhD thesis, Roma Tre University, 2005

work page 2005

[7] [7]

Dostoglou and D

S. Dostoglou and D. Salamon. Self-Dual Instantons and Holomorphic Curves . Annals of Mathematics , 139(3):581--640, May 1994

work page 1994

[8] [8]

Hakan Eliasson, Benoît Grébert, and Sergeï B

L. Hakan Eliasson, Benoît Grébert, and Sergeï B. Kuksin. KAM for the non-linear B eam equation 2: A normal form theorem , 2015

work page 2015