Existence of solitary waves in dipolar quantum gases

Antonelli, Paolo; Sparber, Christof

doi:10.1016/j.physd.2010.10.004

Cited by 41 publications

(88 citation statements)

References 29 publications

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“…Note that there is no contradiction with [5, Proposition 4.1 and Lemma 5.1], since the solitary wave constructed in [1] corresponds to an initial data with a positive energy, while finite time blow-up is established in [5] only for negative energy solutions. A complete analysis of such situations has been done in [11,12].…”

Section: Introductionmentioning

confidence: 86%

“…In [1], the authors have studied (1.5) (without the term V (x)) by using the first approach. More precisely, they introduced the following minimization problem:…”

Section: Introductionmentioning

confidence: 99%

“…They then deduced the main result of their paper [1, Theorem 1.1], which we recall for the convenience of the reader. Theorem 1.1 (Antonelli-Sparber [1]). Let λ 1 , λ 2 ∈ R be such that the following condition holds:…”

Section: Introductionmentioning

confidence: 99%

“…According to the breakthrough paper of Grillakis, Shatah and Strauss [8], the stable solutions of (1.5) are the ones obtained via the variational problem (1.8). In [1], the authors seem to be very skeptical concerning the use of such approach in this context. In [1, p. 427], after the introduction of the energy functional they stated 'At this point it might be tempting to study (1.5) via minimization of the energy E(u).…”

Section: Introductionmentioning

confidence: 99%

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Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases

Carles

Hajaiej²

2015

Bulletin of the London Mathematical Society

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ABSTRACT. We study the stability of the standing wave solutions of a Gross-Pitaevskii equation describing Bose-Einstein condensation of dipolar quantum gases and characterize their orbit. As an intermediate step, we consider the corresponding constrained minimization problem and establish existence, symmetry and uniqueness of the ground state solutions. INTRODUCTIONSince the experimental realization of the first Bose-Einstein condensate (BEC) by Eric Cornell and Carl Wieman in 1995, tremendous efforts have been undertaken by mathematicians to exploit this achievement especially in atomic physics and optics. In the last years, a new kind of quantum gases with dipolar interaction, which acts between particles as a permanent magnetic or electric dipole moment has attracted the attention of a lot of scientists. The interactions between particles are both long-range and non-isotropic. Describing the corresponding BEC via Gross Pitaevskii approximation, one gets the following nonlinear Schrödinger equation (1.1) i ∂ t ψ = − 2 2m ∆ψ + g|ψ| 2 ψ + d 2 (K * |ψ| 2 )ψ + V (x)ψ, t ∈ R, x ∈ R 3 , where |g| = 4π 2 N |a| m , N ∈ N is the number of particles, m denotes the mass of individual particles and a its corresponding scattering length. The external potential V (x) describes the electromagnetic trap and has the following harmonic confinementThe factor d 2 denotes the strength of the dipole moment in Gaussian units andwhere θ = θ(x) is the angle between x ∈ R 3 and the dipole axis n ∈ R 3 . The local term g|ψ| 2 ψ describes the short-range interaction forces between particles, while the non-local potential K * |ψ| 2 describes their long-range dipolar interactions. For the mathematical analysis, it is more convenient to rescale (1.1) into the following dimensionless formThis work was supported by the French ANR projects SchEq (ANR-12-JS01-0005-01) and BoND (ANR-13-BS01-0009-01). , and a 0 = m .In the following, we assume that λ 1 and λ 2 are two given real-valued parameters.In [5], the authors have studied the existence and uniqueness of the equation (1.3) with initial condition ψ 0 ∈ H 1 (R 3 ),They have established that (1.4) has a unique, global solution if λ 1 4 3 πλ 2 0. They called this situation stable regime, referring to the fact that no singularity in formed in finite time. In this paper, we study another notion of stability, that is, the stability of standing waves. They have also showed that in the unstable regime (λ 1 < 4 3 πλ 2 ), finite time blow up may occur, hence the denomination.The evidence of blow-up relies on a function for which the corresponding energy is strictly negative ([5, Lemma 5.1]). They concluded using the virial approach of Zakharov and Glassey. Some refinements of the above result have been discussed in [5, Proposition 5.4].The most important issue in view of the applications of (1.4) in atomic physics and quantum optics seems to be the study of ground state solutions of (1.4). These solutions are the "only" observable states in experiments. A standing wave solution of (1.4) is a wave func...

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Section: Introductionmentioning

confidence: 86%

“…In [1], the authors have studied (1.5) (without the term V (x)) by using the first approach. More precisely, they introduced the following minimization problem:…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases

Carles

Hajaiej²

2015

Bulletin of the London Mathematical Society

View full text Add to dashboard Cite

show abstract

“…Bao, Abdallah and Cai established the existence and uniqueness of the ground state and well‐posedness of the Cauchy problems associated to the quasi‐2D equation as well as quasi‐1D equation through dimension reduction(see also []). Antonelli and Sparber proved the existence of standing waves in the mean field description of dipolar quantum gases, which provides a strict argument for the possible existence of solitary waves in Bose‐Einstein condensation. Moreover, by constructing special cross‐constrained invariant sets, Ma and Wang have obtained threshold of global existence and finite time blowup of solutions for Equation .…”

Section: Introductionmentioning

confidence: 98%