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On the Convexity of the KdV Hamiltonian


Kappeler, Thomas; Maspero, Alberto; Molnar, Jan; Topalov, Peter J (2016). On the Convexity of the KdV Hamiltonian. Communications in Mathematical Physics, 346(1):191-236.

Abstract

Motivated by perturbation theory, we prove that the nonlinear part $ \mathit{H^*}$ of the KdV Hamiltonian $\mathit{H^kdv}$, when expressed in action variables $\mathit{I}=(\mathit{I_n})_{\mathit{n}\geqslant1}$, extends to a real analytic function on the positive quadrant $\ell^2_+(\mathbb{N})$ of $\ell^2(\mathbb{N})$ and is strictly concave near 0. As a consequence, the differential of $ \mathit{H^*}$ defines a local diffeomorphism near 0 of $\ell^2_\mathbb{C}(\mathbb{N})$. Furthermore, we prove that the Fourier-Lebesgue spaces $\mathcal{FL}^\mathit{s,p}$ with −1/2 $\leqslant \mathit {s} \leqslant$ 0 and 2 $\leqslant \mathit{p} < \infty$, admit global KdV-Birkhoff coordinates. In particular, it means that $\ell^2_+(\mathbb{N})$ is the space of action variables of the underlying phase space $\mathcal{FL}^{-1/2,4}$ and that the KdV equation is globally in time $\mathit{C}^0$-well-posed on $\mathcal{FL}^{-1/2,4}$.

Abstract

Motivated by perturbation theory, we prove that the nonlinear part $ \mathit{H^*}$ of the KdV Hamiltonian $\mathit{H^kdv}$, when expressed in action variables $\mathit{I}=(\mathit{I_n})_{\mathit{n}\geqslant1}$, extends to a real analytic function on the positive quadrant $\ell^2_+(\mathbb{N})$ of $\ell^2(\mathbb{N})$ and is strictly concave near 0. As a consequence, the differential of $ \mathit{H^*}$ defines a local diffeomorphism near 0 of $\ell^2_\mathbb{C}(\mathbb{N})$. Furthermore, we prove that the Fourier-Lebesgue spaces $\mathcal{FL}^\mathit{s,p}$ with −1/2 $\leqslant \mathit {s} \leqslant$ 0 and 2 $\leqslant \mathit{p} < \infty$, admit global KdV-Birkhoff coordinates. In particular, it means that $\ell^2_+(\mathbb{N})$ is the space of action variables of the underlying phase space $\mathcal{FL}^{-1/2,4}$ and that the KdV equation is globally in time $\mathit{C}^0$-well-posed on $\mathcal{FL}^{-1/2,4}$.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:January 2016
Deposited On:01 Feb 2017 11:32
Last Modified:29 Mar 2017 07:14
Publisher:Springer
ISSN:0010-3616
Publisher DOI:https://doi.org/10.1007/s00220-015-2563-x

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