On the Well-Posedness of the Defocusing mKdV Equation Below $L^{2}$

Kappeler, Thomas; Molnar, Jan-Cornelius

doi:10.1137/16m1096979

Cited by 21 publications

(49 citation statements)

References 17 publications

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“…Regarding the local-in-time analysis, Kappeler-Molnar [19] proved the local wellposedness of the real-valued defocusing mKdV in FL s,p (T) for s ≥ 0 and 1 ≤ p < ∞, where their solutions are understood as the unique limit of classical solutions as in [21]. In view of the scaling critical regularity, this result is almost critical, in the scale of the Fourier-Lebesgue spaces.…”

Section: Andreia Chapoutomentioning

confidence: 97%

“…In view of the scaling critical regularity, this result is almost critical, in the scale of the Fourier-Lebesgue spaces. Unlike the L 2 (T) solutions of [21,26], the solutions in [19] are not yet known to satisfy the equation in the distributional sense. Now, we turn our attention to the global aspect of well-posedness.…”

Section: Andreia Chapoutomentioning

confidence: 99%

“…This result was extended to H s (T) for s ≥ 0 for the real-valued defocusing mKdV by Kappeler-Topalov [21], using the complete integrability of the equation. In [19], Kappeler-Molnar proved global-in-time existence of solutions for the real-valued mKdV with small initial data in FL s,p (T), s ≥ 0 and 1 ≤ p < ∞.…”

Section: Andreia Chapoutomentioning

confidence: 99%

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A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces

Chapouto¹

2021

DCDS

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We study the complex-valued modified Korteweg-de Vries equation (mKdV) on the circle. We first consider the real-valued setting and show global well-posedness of the (usual) renormalized mKdV equation in the Fourier-Lebesgue spaces. In the complex-valued setting, we observe that the momentum plays an important role in the well-posedness theory. In particular, we prove that the complex-valued mKdV equation is ill-posed in the sense of non-existence of solutions when the momentum is infinite, in the spirit of the work on the nonlinear Schrödinger equation by Guo-Oh (2018). This nonexistence result motivates the introduction of the second renormalized mKdV equation, which we propose as the correct model in the complex-valued setting outside H 1 2 (T). Furthermore, imposing a new notion of finite momentum for the initial data, at low regularity, we show existence of solutions to the complex-valued mKdV equation. In particular, we require an energy estimate, from which conservation of momentum follows.2020 Mathematics Subject Classification. 35Q53.

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Section: Andreia Chapoutomentioning

confidence: 97%

Section: Andreia Chapoutomentioning

confidence: 99%

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A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces

Chapouto¹

2021

DCDS

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“…In [5] a result about the existence of a solution that belongs to almost this class is proved. More recently, several papers have appeared on uniform estimates in the Sobolev class, see [18] and [20] (and also [21]). These results miss the case δ x=0 , critical for the scaling, which is an important example for several reasons.…”

Section: Introductionmentioning

confidence: 99%

Singularity formation for the 1-D cubic NLS and the Schrödinger map on <inline-formula><tex-math id="M1">\begin{document}$\mathbb S^2$\end{document}</tex-math></inline-formula>

Banica¹,

Vega

2018

Communications on Pure &Amp; Applied Analysis

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In this note we consider the 1-D cubic Schrödinger equation with data given as small perturbations of a Dirac-δ function and some other related equations. We first recall that although the problem for this type of data is ill-posed one can use the geometric framework of the Schrödinger map to define the solution beyond the singularity time. Then, we find some natural and well defined geometric quantities that are not regular at time zero. Finally, we make a link between these results and some known phenomena in fluid mechanics that inspired this note.2000 Mathematics Subject Classification. Primary: 76B47, 35Q55; Secondary: 35BXX.

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“…e.g. [11,15,16]). In the generality considered, Σ spec(Lu) might have singularities and hence is no longer a Riemann surface.…”

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confidence: 99%

On nomalized differentials on spectral curves associated with the sinh-Gordon equation

Kappeler¹,

Widmer²

2021

JGM

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The spectral curve associated with the sinh-Gordon equation on the torus is defined in terms of the spectrum of the Lax operator appearing in the Lax pair formulation of the equation. If the spectrum is simple, it is an open Riemann surface of infinite genus. In this paper we construct normalized differentials on this curve and derive estimates for the location of their zeroes, needed for the construction of angle variables.

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On the Well-Posedness of the Defocusing mKdV Equation Below $L^{2}$

Cited by 21 publications

References 17 publications

A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces

A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces

Singularity formation for the 1-D cubic NLS and the Schrödinger map on <inline-formula><tex-math id="M1">\begin{document}$\mathbb S^2$\end{document}</tex-math></inline-formula>

On nomalized differentials on spectral curves associated with the sinh-Gordon equation

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