Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation

Béthuel, Fabrice; Gravejat, Philippe; Smets, Didier

doi:10.24033/asens.2271

Cited by 32 publications

(87 citation statements)

References 40 publications

(123 reference statements)

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“…In [5], we relied on this second strategy in order to prove the asymptotic stability of the dark solitons for the Gross-Pitaevskii equation. As mentioned previously, this was performed in the hydrodynamical setting, so that we were not able to handle with the black soliton.…”

Section: Remarkmentioning

confidence: 99%

“…The quantity I R corresponds in rough terms to the amount of momentum starting at a distance R to the right of the soliton. The fact that it is almost increasing in time, as expressed in the following proposition, is a key ingredient for our subsequent analysis (see [5] for an informal discussion regarding the physical interpretation of such a monotonicity).…”

Section: Localization and Smoothness Of The Limit Profilementioning

confidence: 99%

“…Proposition 9 [5]. Let λ ∈ R, and consider a solution u ∈ C 0 (R, L 2 (R)) to the linear Schrödinger equation…”

Section: Localization and Smoothness Of The Limit Profilementioning

confidence: 99%

“…The smoothing properties in Proposition 9 were analysed in a more general context in [19]. We refer the reader to [5] for a detailed proof of this proposition.…”

Section: Localization and Smoothness Of The Limit Profilementioning

confidence: 99%

“…The orbital stability of dark solitons was derived in [28] (see also [1]), whereas the case of the black soliton was solved in [3,22]. More recently, the asymptotic stability of dark solitons was proved in [5]. Concerning chains of solitons, their orbital stability was established in [4], when the solitons in the chain have non-zero speed, are well-separated at initial time, and are ordered according to their speed.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic stability of the black soliton for the Gross–Pitaevskii equation

Gravejat¹,

Smets²

2015

Proc. London Math. Soc.

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Section: Remarkmentioning

confidence: 99%

Section: Localization and Smoothness Of The Limit Profilementioning

confidence: 99%

“…Proposition 9 [5]. Let λ ∈ R, and consider a solution u ∈ C 0 (R, L 2 (R)) to the linear Schrödinger equation…”

Section: Localization and Smoothness Of The Limit Profilementioning

confidence: 99%

“…The smoothing properties in Proposition 9 were analysed in a more general context in [19]. We refer the reader to [5] for a detailed proof of this proposition.…”

Section: Localization and Smoothness Of The Limit Profilementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Asymptotic stability of the black soliton for the Gross–Pitaevskii equation

Gravejat¹,

Smets²

2015

Proc. London Math. Soc.

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Asymptotic Stability of High-dimensional Zakharov–Kuznetsov Solitons

Côte

Muñoz

Pilod

et al. 2015

Arch Rational Mech Anal

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Abstract. We prove that solitons (or solitary waves) of the Zakharov-Kuznetsov (ZK) equation, a physically relevant high dimensional generalization of the Korteweg-de Vries (KdV) equation appearing in Plasma Physics, and having mixed KdV and nonlinear Schrödinger (NLS) dynamics, are strongly asymptotically stable in the energy space. We also prove that the sum of well-arranged solitons is stable in the same space. Orbital stability of ZK solitons is well-known since the work of de Bouard [10]. Our proofs follow the ideas by Martel [30] and Martel and Merle [35], applied for generalized KdV equations in one dimension. In particular, we extend to the high dimensional case several monotonicity properties for suitable half-portions of mass and energy; we also prove a new Liouville type property that characterizes ZK solitons, and a key Virial identity for the linear and nonlinear part of the ZK dynamics, obtained independently of the mixed KdV-NLS dynamics. This last Virial identity relies on a simple sign condition, which is numerically tested for the two and three dimensional cases, with no additional spectral assumptions required. Possible extensions to higher dimensions and different nonlinearities could be obtained after a suitable local well-posedness theory in the energy space, and the verification of a corresponding sign condition.

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On the Asymptotic Stability of $${N}$$ N -Soliton Solutions of the Defocusing Nonlinear Schrödinger Equation

Cuccagna

Jenkins

2016

Commun. Math. Phys.

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We consider the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation for finite density type initial data. Using the ∂ generalization of the nonlinear steepest descent method of Deift and Zhou we derive the leading order approximation to the solution of NLS for large times in the solitonic region of space-time, |x| < 2t, and we provide bounds for the error which decay as t → ∞ for a general class of initial data whose difference from the non vanishing background possesses a fixed number of finite moments and derivatives. Using properties of the scattering map of NLS we derive, as a corollary, an asymptotic stability result for initial data which are sufficiently close to the N -dark soliton solutions of NLS.

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Asymptotic stability in the energy space for dark solitons of the Gross-Pitaevskii equation

Cited by 32 publications

References 40 publications

Asymptotic stability of the black soliton for the Gross–Pitaevskii equation

Asymptotic stability of the black soliton for the Gross–Pitaevskii equation

Asymptotic Stability of High-dimensional Zakharov–Kuznetsov Solitons

On the Asymptotic Stability of $${N}$$ N -Soliton Solutions of the Defocusing Nonlinear Schrödinger Equation

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