On the Convexity of the KdV Hamiltonian

Kappeler, Thomas; Maspero, Alberto; Molnar, Jan-Cornelius; Topalov, Peter

doi:10.1007/s00220-015-2563-x

Cited by 12 publications

(28 citation statements)

References 19 publications

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“…The proof of item (i) can be found in [14,Lemma 4.3]. To prove item (ii) we note that both F n and the canonical root admit opposite signs on opposite sides of G n and they vanish on U n only at λ ± n hence the quotient…”

Section: Roots and Abelian Integralsmentioning

confidence: 99%

“…One application concerns convexity properties of the KdV Hamiltonian. Recall that in [14,Theorem 1] we proved a conjecture of Korotyaev & Kuksin [26] saying that the Hamiltonian H 1 admits a real analytic extension to 2 + and that d I H 1 I=0 = −6Id. It implies that H 1 is strictly concave near I = 0.…”

Section: ) Nmentioning

confidence: 99%

“…In this section we review results from [14,16,17,18,19,21,28]. In addition, we prove asymptotics of spectral quantities for potentials in Fourier Lebesgue spaces.…”

Section: Preliminariesmentioning

confidence: 99%

“…(4). According to [14,15] for q ∈ W s,p the estimates of the periodic eigenvalues (13) can be refined to…”

Section: Asymptotics Of Spectral Quantities In Fourier Lebesgue Spacesmentioning

confidence: 99%

“…It was further shown in [14] that the KdV equation is also globally C 0 -wellposed in the Fourier Lebesgue spaces (ii) For any −2/3 s < −1/2 and t > 0, the solution map…”

Section: Wellposednessmentioning

confidence: 99%

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On the wellposedness of the KdV/KdV2 equations and their frequency maps

Kappeler¹,

Molnar²

2018

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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In form of a case study for the KdV and the KdV2 equations, we present a novel approach of representing the frequencies of integrable PDEs which allows to extend them analytically to spaces of low regularity and to study their asymptotics. Applications include convexity properties of the Hamiltonians and wellposedness results in spaces of low regularity. In particular, it is proved that on H s the KdV2 equation is C 0 -wellposed if s 0 and illposed (in a strong sense) if s < 0.

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Section: Roots and Abelian Integralsmentioning

confidence: 99%

Section: ) Nmentioning

confidence: 99%

“…In this section we review results from [14,16,17,18,19,21,28]. In addition, we prove asymptotics of spectral quantities for potentials in Fourier Lebesgue spaces.…”

Section: Preliminariesmentioning

confidence: 99%

“…(4). According to [14,15] for q ∈ W s,p the estimates of the periodic eigenvalues (13) can be refined to…”

Section: Asymptotics Of Spectral Quantities In Fourier Lebesgue Spacesmentioning

confidence: 99%

“…It was further shown in [14] that the KdV equation is also globally C 0 -wellposed in the Fourier Lebesgue spaces (ii) For any −2/3 s < −1/2 and t > 0, the solution map…”

Section: Wellposednessmentioning

confidence: 99%

See 3 more Smart Citations

On the wellposedness of the KdV/KdV2 equations and their frequency maps

Kappeler¹,

Molnar²

2018

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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On the wellposedness of the KdV equation on the space of pseudomeasures

Kappeler

Molnar

2017

Sel. Math. New Ser.

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In this paper we prove a wellposedness result of the KdV equation on the space of periodic pseudo-measures, also referred to as the Fourier Lebesgue space Fℓ ∞ (T, R), where Fℓ ∞ (T, R) is endowed with the weak* topology. Actually, it holds on any weighted Fourier Lebesgue space Fℓ s,∞ (T, R) with −1/2 < s 0 and improves on a wellposedness result of Bourgain for small Borel measures as initial data. A key ingredient of the proof is a characterization for a distribution q in the Sobolev space H −1 (T, R) to be in Fℓ ∞ (T, R) in terms of asymptotic behavior of spectral quantities of the Hill operator −∂ 2 x +q. In addition, wellposedness results for the KdV equation on the Wiener algebra are proved.

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Reducibility for a fast-driven linear Klein–Gordon equation

Franzoi

Maspero

2019

Annali di Matematica

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We prove a reducibility result for a linear Klein-Gordon equation with a quasi-periodic driving on a compact interval with Dirichlet boundary conditions. No assumptions are made on the size of the driving, however we require it to be fast oscillating. In particular, provided that the external frequency is sufficiently large and chosen from a Cantor set of large measure, the original equation is conjugated to a time independent, diagonal one. We achieve this result in two steps. First, we perform a preliminary transformation, adapted to fast oscillating systems, which moves the original equation in a perturbative setting. Then we show that this new equation can be put to constant coefficients by applying a KAM reducibility scheme, whose convergence requires a new type of Melnikov conditions.

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On the Convexity of the KdV Hamiltonian

Abstract: We prove that the nonlinear part H * of the KdV Hamiltonian H kdv , when expressed in action variables I = (In) n 1 , extends to a real analytic function on the positive quadrant

Cited by 12 publications

References 19 publications

On the wellposedness of the KdV/KdV2 equations and their frequency maps

On the wellposedness of the KdV/KdV2 equations and their frequency maps

On the wellposedness of the KdV equation on the space of pseudomeasures

Reducibility for a fast-driven linear Klein–Gordon equation

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