Motions in a Bose condensate. III. The structure and effective masses of charged and uncharged impurities

We study the local dynamics near general unstable traveling waves of the 3D Gross-Pitaevskii equation in the energy space by constructing smooth local invariant center-stable, center-unstable and center manifolds. We also prove that (i) the center-unstable manifold attracts nearby orbits exponentially before they get away from the traveling waves along the center directions and (ii) if an initial data is not on the center-stable manifolds, then the forward flow will be ejected away from traveling waves exponentially fast. Furthermore, under a non-degenerate assumption, we show the orbital stability of the traveling waves on the center manifolds, which also implies the local uniqueness of the local invariant manifolds. Our approach based on a geometric bundle coordinates should work for a general class of Hamiltonian PDEs.

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“…21) and(5.24), we obtain that R h , R Ψ → 0 as W X 1 ,Q → 0. Consequently Dh cu (W 0 ) = H (W 0 ) and DΨ(t ,W 0 ) = Ψ 1 (t ,W 0 ).…”

mentioning

confidence: 78%

Invariant Manifolds of Traveling Waves of the 3D Gross–Pitaevskii Equation in the Energy Space

Jin

Lin

Zeng

2018

Commun. Math. Phys.

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“…However, they allow us to track the universal mechanical aspects of the vortex-matter interaction, taking place in idealized quantum mixtures. Similar models were used in the past to understand qualitatively the behavior of electron bubbles in liquid helium [22,24]. Various extensions can be incorporated into the formalism, if necessary, to investigate the effects caused by the nonlocal interactions of particles and finite temperatures.…”

Section: Theorymentioning

confidence: 99%

Static and dynamic properties of heavily doped quantum vortices

Pshenichnyuk

2017

New J. Phys.

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Quantum vortices in superfluids may capture matter and deposit it inside their core. By doping vortices with foreign particles one can effectively visualize them and study them experimentally. To acquire a better understanding of the interaction between quantum vortices and matter, and clarify the details of recent experiments, the properties of doped vortices are investigated here theoretically in the regimes where the doping mass becomes close to the total mass of superfluid particles forming a vortex. Such formations are dynamically stable and, possessing both vorticity and enhanced inertia, demonstrate properties that are different from the pure vortex case. The goal of this paper is to define and investigate the universal aspects of heavily doped vortex behavior, which can be realized in different types of quantum mixtures. The proposed 3D model is based on a system of coupled semiclassical matter wave equations that are solved numerically in a wide range of physical parameters. The size, geometry and binding energy of dopants in different regimes are discussed. The coupled motion of a vortex-dopant complex and decoupling conditions are studied. The reconnection of vortices, taken as an example of a fundamental process responsible for the evolution of a quantum turbulent state, is modeled to illustrate the difference between the light and heavy doping cases.Illuminating experiments in quantum turbulence were performed recently using superfluid helium nanodroplets [13][14][15][16]. Vortex filaments in rotating droplets were doped with Ag and Xe atoms and studied using a femtosecond x-ray coherent diffractive imaging technique as well as electron microscopy preceded by surfacedeposition of the samples. Several questions were raised in these works, including the origin of unusual shapes of doped helium droplets and distribution of vortices inside the droplet. Although certain aspects were clarified by theorists [17][18][19], the connection between the rotational motion of a droplet and a dopant still remains unclear [16]. In nanodroplet experiments doping particles are approximately 33 times heavier than fluid atoms and the diameter of each particle is comparable with a vortex core size in helium. This is quite opposite, for instance, in comparison to large and light electron bubbles often used as dopants and well studied in the past both experimentally and theoretically [20][21][22]. Being captured, heavy particles may influence the vortex motion significantly and theoretical modeling is necessary to understand the details of their behavior.According to the results of Gordon and colleagues [8,9], guest atoms in helium above a critical temperature tend to form spherical clusters. In superfluid helium below critical temperature impurities with a certain probability stick together to produce long cylindrical filaments, which are attributed to the presence of quantum vortices. These filaments are stable enough to exist independently, by decoupling from vortices, which allows them to be studied using a surface-depo...

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“…In a series of papers (see, e.g., [GR74], [JR82], [JPR86], [BaMa88], [BGMP89], [PR91], [PN93], [KR95a], [KR95b]), particular attention has been paid to a special class of solutions of (1.1), namely the traveling waves. These are solutions of the form Φ(x, t) = ψ(x − cty), where y ∈ S N −1 is the direction of propagation and c ∈ R * is the speed of the traveling wave.…”

Section: Using the Madelung Transformation φ(X T) = ρ(X T)ementioning

confidence: 99%

“…(1.1) is an extremely active research field at the moment. This equation, with the considered non-zero conditions at infinity, is relevant in a large variety of physical problems such as superconductivity, superfluidity in Helium II, phase transitions and Bose-Einstein condensates ( [BGMP89], [BaMa88], [Be08], [Co98], [GR74], [Gro63], [IoSm78], [JR82], [JPR86]). In nonlinear optics, it appears in the context of dark solitons, that is, localized waves which exist on a stable continuous background ([KiLD98], [KPS95]).…”

Section: Introductionmentioning

confidence: 99%

Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity: some results and open problems

Maris¹

2010

Journées Équations Aux Dérivées Partielles

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Motions in a Bose condensate. III. The structure and effective masses of charged and uncharged impurities

Cited by 27 publications

References 9 publications

Invariant Manifolds of Traveling Waves of the 3D Gross–Pitaevskii Equation in the Energy Space

Invariant Manifolds of Traveling Waves of the 3D Gross–Pitaevskii Equation in the Energy Space

Static and dynamic properties of heavily doped quantum vortices

Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity: some results and open problems

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