Global well-posedness of the derivative nonlinear Schrödinger equation with periodic boundary condition in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:mfrac></mml:mrow></mml:msup></mml:math>

Mosincat, Razvan

doi:10.1016/j.jde.2017.05.026

Cited by 21 publications

(11 citation statements)

References 44 publications

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“…Wu [75,76] showed that it is globally well-posed in H 1 (R) under the condition u 0 L 2 < 2 √ π. Guo and Wu [21] later proved that it is globally well-posed in H 1 2 (R) under the same condition of initial data, see also [9,10,61] for the previous results on the low regularity. The same results also hold in the periodic case, see Mosincat and Oh [59] in H 1 (T), and Mosincat [58] in H 1 2 (T). More recently, Jenkins, Liu, Perry and Sulem [38] proved that the Cauchy problem (1.2) is globally well-posed in the weighted Sobolev space H 2,2 (R).…”

Section: Introductionsupporting

confidence: 62%

Optimal small data Scattering for the generalized derivative nonlinear Schrödinger equations

Bai,

Wu,

Xue

2018

Preprint

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In this work, we consider the following generalized derivative nonlinear Schrödinger equationsWe prove that when σ ≥ 2, the solution is global and scattering when the initial data is small in H s (R), s ≥ 1 2 . Moreover, we show that when 0 < σ < 2, there exist a class of solitary wave solutions {φ c } satisfywhen c tends to some endpoint, which is against the small data scattering statement. Therefore, the restriction σ ≥ 2 is optimal for scattering.

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Section: Introductionsupporting

confidence: 62%

Optimal small data Scattering for the generalized derivative nonlinear Schrödinger equations

Bai,

Wu,

Xue

2018

Preprint

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“…The highest value of the mass for which global existence in H 1 (T) holds is δ = 2 π/|β|. This was shown for the non-periodic framework in [Wu13] and the argument was adapted to periodic DNLS in [MO15] (similarly, for the best result in H 1 2 (T) see [Mos17]). Existence of global solution of DNLS on R without any condition on the mass has been proven by inverse scattering method in [LPS16].…”

Section: Introductionmentioning

confidence: 79%

Invariant measures for the periodic derivative nonlinear Schrödinger equation

2018

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We construct invariant measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation (DNLS) for small data in L 2 and we show these measures to be absolutely continuous with respect to the Gaussian measure. The key ingredient of the proof is the analysis of the gauge group of transformations associated to DNLS. As an intermediate step for our main result, we prove quasi-invariance with respect to the gauge maps of the Gaussian measure on L 2 with covariance (I + (−∆) k ) −1 for any k 2.

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“…The result in [49] was extended to the periodic setting in [35]. Finally, the argument in [5] was further advanced in [12,34] to treat the endpoint case s = 1 2 and M (q) < 4π; see also [47] for earlier work in the periodic setting.…”

Section: Introductionmentioning

confidence: 99%

On the well-posedness problem for the derivative nonlinear Schrödinger equation

Killip¹,

Ntekoume²,

Vişan³

2021

Preprint

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We consider the derivative nonlinear Schrödinger equation in one space dimension, posed both on the line and on the circle. This model is known to be completely integrable and L 2 -critical with respect to scaling.The first question we discuss is whether ensembles of orbits with L 2equicontinuous initial data remain equicontinuous under evolution. We prove that this is true under the restriction M (q) = |q| 2 < 4π. We conjecture that this restriction is unnecessary.Further, we prove that the problem is globally well-posed for initial data in H 1/6 under the same restriction on M . Moreover, we show that this restriction would be removed by a successful resolution of our equicontinuity conjecture.

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Global well-posedness of the derivative nonlinear Schrödinger equation with periodic boundary condition inH12

Cited by 21 publications

References 44 publications

Optimal small data Scattering for the generalized derivative nonlinear Schrödinger equations

Optimal small data Scattering for the generalized derivative nonlinear Schrödinger equations

Invariant measures for the periodic derivative nonlinear Schrödinger equation

On the well-posedness problem for the derivative nonlinear Schrödinger equation

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