Smooth branch of travelling waves for the Gross-Pitaevskii equation in R2 for small speed

Chiron, David; Pacherie, Eliot

doi:10.2422/2036-2145.201906_015

Cited by 10 publications

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“…In that limit, we show the uniqueness of this minimizer, up to the invariances of the problem, hence proving the orbital stability of this travelling wave. This work is a follow up to two previous papers [16], [15], where we constructed and studied a particular travelling wave of the equation. We show a uniqueness result on this travelling wave in a class of functions that contains in particular all possible minimizers of the energy.Besides the energy, the momentum is another quantity formally conserved by the (NLS) flow, and is associated with the invariance by translation of (NLS).…”

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confidence: 87%

“…For instance, we may use variational methods, such as a mountain pass argument in [11] and in [3], or by minimizing the energy at fixed kinetic energy ( [7], [14]). Also, we have constructed in [16] a travelling wave by perturbative methods, taking for ansatz a pair of vortices, by following the Lyapounov-Schmidt reduction method as initiated in [21]. Vortices are stationary solutions of (NLS) of degrees n ∈ Z * (see [26], [40], [46], [29], [13]):…”

Section: A Smooth Branch Of Travelling Waves For Large Momentummentioning

confidence: 99%

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A uniqueness result for the two vortex travelling wave in the Nonlinear Schrodinger equation

Chiron¹,

Pacherie²

2021

Preprint

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For the Nonlinear Schrödinger equation in dimension 2, the existence of a global minimizer of the energy at fixed momentum has been established by Bethuel-Gravejat-Saut [7] (see also [14]). This minimizer is a travelling wave for the Nonlinear Schrödinger equation. For large momentums, the propagation speed is small and the minimizer behaves like two well separated vortices. In that limit, we show the uniqueness of this minimizer, up to the invariances of the problem, hence proving the orbital stability of this travelling wave. This work is a follow up to two previous papers [16], [15], where we constructed and studied a particular travelling wave of the equation. We show a uniqueness result on this travelling wave in a class of functions that contains in particular all possible minimizers of the energy.Besides the energy, the momentum is another quantity formally conserved by the (NLS) flow, and is associated with the invariance by translation of (NLS). Formally, the momentum of u is 1 2 Ê 2 Re(i∇uū) ∈ Ê 2 , but its precise definition requires some care in the energy space due to the condition at infinity (see [37] in dimension larger than two and [14] ) for instance, then the expression of the momentum reduces toRe i∇u(ū − 1) .In addition to the translation invariance, the (NLS) equation is also phase shift invariant, that is invariant by multiplication by a complex of modulus one, and rotation invariant. Travelling waves for (NLS)Following the works in the physical literature of Jones and Roberts (see [32], [31]), there has been a large amount of mathematical works on the question of existence and properties of travelling waves solutions in the (NLS) equation, that is solution offor some c > 0, corresponding to particular solutions of (NLS) of the form Ψ(t, x) = u(x 1 , x 2 + ct) (due to the rotational invariance, we may always assume that the traveling wave moves along the direction − − → e 2 ). We refer to [6] for an overview on these problems in several dimensions. A natural approach is to look at the minimizing problem for p > 0It was shown by Bethuel-Gravejat-Saut in [7] that there exists a minimizer to this problem.Theorem 1.2 ([7]) For any p > 0, there exists a non constant function u p ∈ E and c(u p ) > 0 such that P 2 (u p ) = p, u p is a solution to (TW c(up) )(u p ) = 0 and E(u p ) = E min (p).Furthermore, any minimizer for E min (p) is, up to a translation in x 1 , even in x 1 .The strategy is to look at the corresponding minimization problem on tori (this avoids the problems with the definition of the momentum) larger and larger, and then pass to the limit. For the minimizing problem E min (p), the compactness of minimizing sequences has been shown later on in [14] for the natural semi-distance onTheorem 1.3 ([14]) For any p > 0, and any minimizing sequence (u n ) n∈AE for E min (p), there exists a subsequence (u nj ) j∈AE , a sequence of translations (y j ) j∈AE and a non constant function u p ∈ E such that D 0 (u nj , u p ) → 0, P 2 (u nj ) → P 2 (u p ) = p and E(u nj ) → E(u p ) = E min (p) as ...

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confidence: 87%

Section: A Smooth Branch Of Travelling Waves For Large Momentummentioning

confidence: 99%

A uniqueness result for the two vortex travelling wave in the Nonlinear Schrodinger equation

Chiron¹,

Pacherie²

2021

Preprint

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“…and iV 1 (see e.g. [7,Proposition 1.3] and also the Fredholm alternative in [10,Theorem 2]). These estimates however do not allow to control the nonlinear terms arising in the expansion of the renormalized Ginzburg-Landau energy, and it does not seem possible to derive nonlinear stability of V 1 based (exclusively) on the linear analysis of B.…”

Section: Concerning Coercivity and Theoremmentioning

confidence: 99%

“…The question whether V 1 is stable as a stationary solution to (3), and not only orbitally stable, is still open. There is no immediate obstruction to that stronger form of stability since, although there exist travelling waves of (3) with arbitrarily small speed (see [4,8]), they are not small perturbations of the vortex but instead perturbations of a vortex-antivortex pair, and have finite Ginzburg-Landau energy.…”

Section: Introductionmentioning

confidence: 99%

On the stability of the Ginzburg-Landau vortex

Gravejat¹,

Pacherie²,

Smets³

2021

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We introduce a functional framework taylored to investigate the minimality and stability properties of the Ginzburg-Landau vortex of degree one on the whole plane. We prove that a renormalized Ginzburg-Landau energy is well-defined in that framework and that the vortex is its unique global minimizer up to the invariances by translation and phase shift. Our main result is a nonlinear coercivity estimate for the renormalized energy around the vortex, from which we can deduce its orbital stability as a solution to the Gross-Pitaevskii equation, the natural Hamiltonian evolution equation associated to the Ginzburg-Landau energy.

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“…A very active field of research is the study of the location and dynamics of vortices, namely, the zeroes of the wave function ψ. The existence of multivortices traveling waves with small speed has been proved in dimension d = 2, see [41,14,15]. In dimension 3 there are traveling vortex rings ( [40]) as well as leapfrogging vortex rings, see [33].…”

Section: Introductionmentioning

confidence: 99%

Finite energy traveling waves for the Gross-Pitaevskii equation in the subsonic regime

Bellazzini¹,

Ruiz²

2019

Preprint

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In this paper we study the existence of finite energy traveling waves for the Gross-Pitaevskii equation. This problem has deserved a lot of attention in the literature, but the existence of solutions in the whole subsonic range was a standing open problem till the work of Mariş in 2013. However, such result is valid only in dimension 3 and higher. In this paper we first prove the existence of finite energy traveling waves for almost every value of the speed in the subsonic range. Our argument works identically well in dimensions 2 and 3.With this result in hand, a compactness argument could fill the range of admissible speeds. We are able to do so in dimension 3, recovering the aforementioned result by Mariş. The planar case turns out to be more difficult and the compactness argument works only under an additional assumption on the vortex set of the approximating solutions. J. B. is partially supported by Project 2016 Dinamica di equazioni nonlineari dispersive of FONDAZIONE DI SARDEGNA. D. R. has been supported by the FEDER-MINECO Grants MTM2015-68210-P and PGC2018-096422-B-I00, and by J. Andalucia (FQM-116).

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Smooth branch of travelling waves for the Gross-Pitaevskii equation in R2 for small speed

Cited by 10 publications

References 0 publications

A uniqueness result for the two vortex travelling wave in the Nonlinear Schrodinger equation

A uniqueness result for the two vortex travelling wave in the Nonlinear Schrodinger equation

On the stability of the Ginzburg-Landau vortex

Finite energy traveling waves for the Gross-Pitaevskii equation in the subsonic regime

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