Global well-posedness for the derivative nonlinear Schrödinger equation in $H^{\frac 12} (\mathbb{R} )$

Guo, Zihua; Wu, Yifei

doi:10.3934/dcds.2017010

Cited by 45 publications

(46 citation statements)

References 17 publications

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“…The IVP associated to (1.5) has been studied in several publications (see for instance [2], [5], [6], [14], [15], [16], [33], [36], [37]) where among other qualitative properties local and global well-posedness issues were investigated. In particular, a global sharp well-posedness result was obtained by Guo and Wu [11] in H 1/2 (R) for initial data satisfying (1.8) u 0 2 2 ≤ 4π, see also [41]. The condition (1.8) guarantees the energy E(·) is positive via sharp Gagliardo-Nirenberg inequality.…”

Section: Introductionmentioning

confidence: 95%

On a Class of Solutions to the Generalized Derivative Schrödinger Equations

Linares

Ponce

Santos

2019

Acta. Math. Sin.-English Ser.

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In this work we shall consider the initial value problem associated to the generalized derivative Schrödinger equations ∂tu = i∂ 2x u + µ |u| α ∂xu, x, t ∈ R, 0 < α ≤ 1 and |µ| = 1, andFollowing the argument introduced by Cazenave and Naumkin [3] we shall establish the local well-posedness for a class of small data in an appropriate weighted Sobolev space. The other main tools in the proof include the homogeneous and inhomogeneous versions of the Kato smoothing effect for the linear Schrödinger equation established by Kenig-Ponce-Vega in [21].

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Section: Introductionmentioning

confidence: 95%

On a Class of Solutions to the Generalized Derivative Schrödinger Equations

Linares

Ponce

Santos

2019

Acta. Math. Sin.-English Ser.

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“…2. Prior to [15], in [40,41], Wu first obtained L 2 -norm threshold improvements for energy-space initial data.…”

Section: Comments and Remarks For Dnls On The Real Line Yin Yin Su Winmentioning

confidence: 99%

Unconditional uniqueness for the derivative nonlinear Schrödinger equation on the real line

Mosincat¹,

Yoon²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We prove the unconditional uniqueness of solutions to the derivative nonlinear Schrödinger equation (DNLS) in an almost end-point regularity. To this purpose, we employ the normal form method and we transform (a gauge-equivalent) DNLS into a new equation (the so-called normal form equation) for which nonlinear estimates can be easily established in H s (R), s > 1 2 , without appealing to an auxiliary function space. Also, we prove that low-regularity solutions of DNLS satisfy the normal form equation and this is done by means of estimates in the H s−1 (R)-norm.

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“…4π (or }u 0 } L 2 " ? 4π with negative momentum), there exists a unique solution u P CpR, H 1{2 pRqq [6,7]. A key structural property of DNLS discovered by Kaup and Newell [11] is that it is integrable by inverse scattering: that is, there is a linear spectral problem with upx, tq as potential whose spectral data (consisting of a reflection coefficient, describing the continuous spectrum of the linear problem, together with eigenvalues and norming constants, describing the discrete spectrum of the linear problem) evolve linearly under the flow.…”

Section: {2mentioning

confidence: 99%