Stability and evolution of the quiescent and travelling solitonic bubbles

Barashenkov, I. V.; Panova, Elena

doi:10.1016/0167-2789(93)90184-3

Cited by 103 publications

(84 citation statements)

References 14 publications

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“…Also, it should be noted that as in this case the dispersion relation implies that ω > k 2 , the allowable region in the (k, ω) plane for dark solitons is located above the parabola ω = 1 2 k 2 . As the integrals of motion of the NLS equation refer to both the background and the dark soliton, the integrals of motion for the dark soliton are renormalized so as to extract the contribution of the background [112][113][114]. In particular, the renormalized momentum and energy of the dark soliton (27) read (for k = 0):…”

Section: Bright and Dark Matter-wave Solitonsmentioning

confidence: 99%

Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques

2008

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Abstract. The aim of the present review is to introduce the reader to some of the physical notions and of the mathematical methods that are relevant to the study of nonlinear waves in Bose-Einstein Condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyze some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons, as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g., the linear or the nonlinear limit, or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools.PACS numbers: 03.75. Kk, 03.75.Lm, 05.45 • EP: Ermakov-Pinney (Equation)• GP: Gross-Pitaevskii (Equation)• KdV: Korteweg-de Vries (Equation)• LS: Lyapunov-Schmidt (Technique)• MT: Magnetic Trap• NLS: Nonlinear Schrödinger (Equation)• NPSE: Non-polynomial Schrödinger Equation IntroductionThe phenomenon of Bose-Einstein condensation is a quantum phase transition originally predicted by Bose [1] and Einstein [2,3] in 1924. In particular, it was shown that below a critical transition temperature T c , a finite fraction of particles of a boson gas (i.e., whose particles obey the Bose statistics) condenses into the same quantum state, known as the Bose-Einstein condensate (BEC). Although BoseEinstein condensation is known to be a fundamental phenomenon, connected, e.g., to superfluidity in liquid helium and superconductivity in metals (see, e.g., Ref.[4]), BECs were experimentally realized 70 years after their theoretical prediction: this major achievement took place in 1995, when different species of dilute alkali vapors confined in a magnetic trap (MT) were cooled down to extremely low temperatures [5][6][7], and has already been recognized through the 2001 Nobel prize in Physics [8,9]. This first unambiguous manifestation of a macroscopic quantum state in a manybody system sparked an explosion of activity, as reflected by the publication of several thousand papers related to BECs since then. Nowadays there exist more than fifty experimental BEC groups around the world, while an enormous amount of theoretical work has followed and driven the experimental efforts, with an impressive impact on many branches of Physics.From a theoretical standpoint, and for experimentally relevant conditions, the static and dynamical properties of a BEC can be described by means of an effective mean-field equation known as the Gross-Pitaevskii (GP) equation [10,11]. This is a variant of the famous nonlinear Schrödinger (NLS) equation [12] (incorporating an external potential used to confine th...

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Section: Bright and Dark Matter-wave Solitonsmentioning

confidence: 99%

Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques

2008

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“…In the case of dark solitons of the form u(x,t) = u(x − V t), with |u| 2 → ρ 0 as |x| → ∞, a similar criterion [25][26][27][28] involves the (renormalized) field momentum…”

Section: U + δU = R[u(xt); Xt]mentioning

confidence: 99%

Refined empirical stability criterion for nonlinear Schrödinger solitons under spatiotemporal forcing

et al. 2011

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We investigate the dynamics of traveling oscillating solitons of the cubic nonlinear Schrödinger (NLS) equation under an external spatiotemporal forcing of the form f (x,t) = a exp[iK(t)x]. For the case of time-independent forcing, a stability criterion for these solitons, which is based on a collective coordinate theory, was recently conjectured. We show that the proposed criterion has a limited applicability and present a refined criterion which is generally applicable, as confirmed by direct simulations. This includes more general situations where K(t) is harmonic or biharmonic, with or without a damping term in the NLS equation. The refined criterion states that the soliton will be unstable if the "stability curve" p(v), where p(t) and v(t) are the normalized momentum and the velocity of the soliton, has a section with a negative slope. In the case of a constant K and zero damping, we use the collective coordinate solutions to compute a "phase portrait" of the soliton where its dynamics is represented by two-dimensional projections of its trajectories in the four-dimensional space of collective coordinates. We conjecture, and confirm by simulations, that the soliton is unstable if a section of the resulting closed curve on the portrait has a negative sense of rotation.

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“…In the latter setting, dark solitons typically move along the localized ground state trapped by the external potentials [42]. The main model for BECs is a modification of the NLS equation (1.1) with an external potential, which is called the Gross-Pitaevskii (GP) equation, 2) where ǫ ∈ R is the strength of the potential V (x) and V (x) : R → R is assumed to be a smooth function satisfying one of the three properties:…”

Section: Introductionmentioning

confidence: 99%

“…Among others, several analytical results were important in the development of dark solitons in recent years: perturbation theory based on renormalized power [24] and momentum [23], orbital stability of dark solitons [1,2], completeness of eigenfunctions in the cubic NLS [8,15], inverse scattering for the vector cubic NLS equation [41], construction of the Evans function for dark solitons in the perturbed cubic NLS [19], asymptotic analysis of the radiation and dynamics of dark solitons [26,27,35], and spectral analysis of transverse instabilities of one-dimensional dark solitons [25,36].…”

Section: Introductionmentioning

confidence: 99%

Dark solitons in external potentials

Pelinovsky

Kevrekidis

2007

Z. angew. Math. Phys.

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We consider the persistence and stability of dark solitons in the Gross-Pitaevskii (GP) equation with a small decaying potential. We show that families of black solitons with zero speed originate from extremal points of an appropriately defined effective potential and persist for sufficiently small strength of the potential. We prove that families at the maximum points are generally unstable with exactly one real positive eigenvalue, while families at the minimum points are generally unstable with exactly two complex-conjugated eigenvalues with positive real part. This mechanism of destabilization of the black soliton is confirmed in numerical approximations of eigenvalues of the linearized GP equation and full numerical simulations of the nonlinear GP equation with cubic nonlinearity. We illustrate the monotonic instability associated with the real eigenvalues and the oscillatory instability associated with the complex eigenvalues and compare the numerical results of evolution of a dark soliton with the predictions of Newton's particle law for its position.

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Stability and evolution of the quiescent and travelling solitonic bubbles

Cited by 103 publications

References 14 publications

Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques

Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques

Refined empirical stability criterion for nonlinear Schrödinger solitons under spatiotemporal forcing

Dark solitons in external potentials

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