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On the Well-posedness of the defocusing mKdV equation below $L^{2}$


Kappeler, Thomas; Molnar, Jan-Cornelius (2017). On the Well-posedness of the defocusing mKdV equation below $L^{2}$. SIAM Journal on Mathematical Analysis, 49(3):2191-2219.

Abstract

We prove that the renormalized defocusing modified KdV (mKdV) equation on the circle is locally in time $C^{0}$-well-posed on the Fourier Lebesgue space ${\mathscr{F}\ell}^p$ for any $2 < p < \infty$. The result implies that the defocusing mKdV equation itself is ill-posed on these spaces since the renormalizing phase factor becomes infinite. The proof is based on the fact that the mKdV equation is an integrable PDE whose Hamiltonian is in the nonlinear Schrödinger hierarchy. A key ingredient is a novel way of representing the bi-infinite sequence of frequencies of the renormalized defocusing mKdV equation, allowing us to analytically extend them to ${\mathscr{F}\ell}^p$ for any $2 \le p < \infty$ and to deduce asymptotics for $n \to \pm \infty$.

Abstract

We prove that the renormalized defocusing modified KdV (mKdV) equation on the circle is locally in time $C^{0}$-well-posed on the Fourier Lebesgue space ${\mathscr{F}\ell}^p$ for any $2 < p < \infty$. The result implies that the defocusing mKdV equation itself is ill-posed on these spaces since the renormalizing phase factor becomes infinite. The proof is based on the fact that the mKdV equation is an integrable PDE whose Hamiltonian is in the nonlinear Schrödinger hierarchy. A key ingredient is a novel way of representing the bi-infinite sequence of frequencies of the renormalized defocusing mKdV equation, allowing us to analytically extend them to ${\mathscr{F}\ell}^p$ for any $2 \le p < \infty$ and to deduce asymptotics for $n \to \pm \infty$.

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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:1 March 2017
Deposited On:18 Jan 2018 10:50
Last Modified:18 Apr 2018 11:49
Publisher:Society for Industrial and Applied Mathematics
ISSN:0036-1410
Funders:Schweizerischer Nationalfonds
OA Status:Green
Publisher DOI:https://doi.org/10.1137/16M1096979

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