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

Gravejat, Philippe; Smets, Didier

doi:10.1112/plms/pdv025

Cited by 34 publications

(65 citation statements)

References 41 publications

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“…Proof. Thanks to the fast (exponential) decay of 1 − u 2 1 − u 2 2 to zero at infinity, the same arguments as in [13] imply that the operator T is compact in H, so that its spectrum in H is purely discrete. Therefore, the spectrum of the operator L − := I − γT in H consists of isolated eigenvalues λ accumulating to the point λ 0 = 1.…”

Section: Coercivity In a Weighted H 1 Spacementioning

confidence: 81%

“…The new metric was introduced uniformly on the real line, so that the compact support controlled by the parameter A > 0 in the family of distances ρ A in (1.6) becomes abundant. Once the coercivity of the energy is obtained in the weighted H 1 space, nonlinear orbital stability and the asymptotic stability of black solitons can be established by available analytical techniques in [13].…”

Section: Introductionmentioning

confidence: 99%

“…The new variables introduced in [13] were further used in analysis of nonlinear orbital stability of black solitons in the H 2 space by using a higher-order energy of the cubic NLS equation [10]. To tackle with the lack of coercivity, the family of distances given by (1.6) was still used and analysis was developed separately inside and outside the compact support.…”

Section: Introductionmentioning

confidence: 99%

“…By Lemma 3.1, the imaginary part of perturbations W to the domain walls U is well controlled in the metric space H subject to the two orthogonality conditions (3.3) that specify complex phases of the two components of Ψ = (ψ 1 , ψ 2 ) due to gauge rotations. Compared to[13], no additional orthogonality conditions are needed because the domain wall solutions are true minima of the energy functional E. Remark 3.2. For the real part of perturbations V to the domain walls U , we can only add one orthogonality condition that specifies a spatial translation of the solution Ψ.…”

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confidence: 99%

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Orbital Stability of Domain Walls in Coupled Gross--Pitaevskii Systems

Contreras¹,

Pelinovsky²,

Plum³

2018

SIAM J. Math. Anal.

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Domain walls are minimizers of energy for coupled one-dimensional Gross-Pitaevskii systems with nontrivial boundary conditions at infinity. It has been shown in [2] that these solutions are orbitally stable in the space of complexḢ 1 functions with the same limits at infinity. In the present work we adopt a new weighted H 1 space to control perturbations of the domain walls and thus to obtain an improved orbital stability result. A major difficulty arises from the degeneracy of linearized operators at the domain walls and the lack of coercivity.(1.7)By Theorem 1.3 in [1], the real minimizers of E satisfying properties (a)-(c) were shown to be the unique real solutions to the system (1.2). By the global well-posedness results in the energy space D in [21], for any Ψ 0 ∈ D ∩ L ∞ (R), there exists a unique global in time solution Ψ(t) ∈ D ∩ L ∞ (R) to the coupled GP system (1.1) with initial data Ψ(0) = Ψ 0 . Moreover, the map t → Ψ(t) is continuous with respect to ρ A and the energy of the coupled GP system (1.1) is preserved along the flow, that is E(Ψ(t)) = E(Ψ 0 ) for all t ∈ R.Finally, by Theorems 1.4 and 1.5 in [2], the following nonlinear orbital stability theorem was established for the domain walls of the coupled GP system (1.1).

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Section: Coercivity In a Weighted H 1 Spacementioning

confidence: 81%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Orbital Stability of Domain Walls in Coupled Gross--Pitaevskii Systems

Contreras¹,

Pelinovsky²,

Plum³

2018

SIAM J. Math. Anal.

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“…Nevertheless orbital stability of the black soliton was proved via variational methods by Bethuel, Gravejat, Saut and Smets [15] and via the inverse scattering transform by Gérard and Zhang [35] (see also [26] for an earlier result and numerical simulations). Recently, Bethuel, Gravejat and Smets proved the orbital stability of a chain of solitons of the Gross-Pitaevskii equation [16] as well as asymptotic stability of the grey solitons [17] and of the black soliton [37]. Existence and stability of traveling waves with a non-zero background for equations of type (1.1) with a general nonlinearity was also studied by Chiron [22,23].…”

Section: Introductionmentioning

confidence: 98%

On the Cauchy Problem and the Black Solitons of a Singularly Perturbed Gross--Pitaevskii Equation

Ianni¹,

Coz²,

Royer³

2017

SIAM J. Math. Anal.

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We consider the one-dimensional Gross-Pitaevskii equation perturbed by a Dirac potential. Using a fine analysis of the properties of the linear propagator, we study the well-posedness of the Cauchy Problem in the energy space of functions with modulus 1 at infinity. Then we show the persistence of the stationary black soliton of the unperturbed problem as a solution. We also prove the existence of another branch of non-trivial stationary waves. Depending on the attractive or repulsive nature of the Dirac perturbation and of the type of stationary solutions, we prove orbital stability via a variational approach, or linear instability via a bifurcation argument.

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

Abstract: We introduce a new framework for the analysis of the stability of solitons for the one-dimensional Gross-Pitaevskii equation. In particular, we establish the asymptotic stability of the black soliton with zero speed.

Cited by 34 publications

References 41 publications

Orbital Stability of Domain Walls in Coupled Gross--Pitaevskii Systems

Orbital Stability of Domain Walls in Coupled Gross--Pitaevskii Systems

On the Cauchy Problem and the Black Solitons of a Singularly Perturbed Gross--Pitaevskii Equation

Asymptotic Stability of High-dimensional Zakharov–Kuznetsov Solitons

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