The Defocusing Nonlinear Schrödinger Equation

Kevrekidis, P. G.; Frantzeskakis, D. J.; Carretero-González, R.

doi:10.1137/1.9781611973945

Cited by 197 publications

(334 citation statements)

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“…Upon suitable standard reductions [1,2], the model can be transformed to its dimensionless version in the form:…”

Section: Theoretical/numerical Resultsmentioning

confidence: 99%

“…For a single-component system, an extensive discussion of the existence and stability of excited states such as dark solitons or vortices can be found in the respective 1d and 2d chapters of [2]. In a two-component system with inter-component repulsion, a dark soliton or a vortex in one component will induce a potential in the second component.…”

Section: Introductionmentioning

confidence: 99%

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Vector dark-antidark solitary waves in multicomponent Bose-Einstein condensates

et al. 2016

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Multi-component Bose-Einstein condensates exhibit an intriguing variety of nonlinear structures.In recent theoretical work, the notion of magnetic solitons has been introduced. Here we generalize this concept to vector dark-antidark solitary waves in multi-component Bose-Einstein condensates.We first provide concrete experimental evidence for such states in an atomic BEC and subsequently illustrate the broader concept of these states, which are based on the interplay between miscibility and inter-component repulsion. Armed with this more general conceptual framework, we expand the notion of such states to higher dimensions presenting the possibility of both vortex-antidark states and ring-antidark-ring (dark soliton) states. We perform numerical continuation studies, investigate the existence of these states and examine their stability using the method of Bogolyubovde Gennes analysis. Dark-antidark and vortex-antidark states are found to be stable for broad parametric regimes. In the case of ring dark solitons, where the single-component ring state is known to be unstable, the vector entity appears to bear a progressively more and more stabilizing role as the inter-component coupling is increased.

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“…Upon suitable standard reductions [1,2], the model can be transformed to its dimensionless version in the form:…”

Section: Theoretical/numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Vector dark-antidark solitary waves in multicomponent Bose-Einstein condensates

et al. 2016

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“…For smaller values of D, higher eigenmodes may become unstable too, and the respective instabilities may eventually (i.e., at large µ + ) even dominate the respective growth rate. The enhanced instability at smaller D is a natural feature to expect: indeed, as D decreases, the notch shrinks, turning into a circular quasi-1D dark soliton, whose snaking instability in two-dimensional settings is well known [8]. It is also relevant to stress that the oscillatory pattern, featured, especially, by the m = 0 mode is associated with the presence of gaps in the spectrum (for our finite-domain computation), which allow the eigenmode to periodically restabilize, before it collides with another one and destabilizes anew.…”

Section: B Numerical Resultsmentioning

confidence: 99%

“…Thus, gradually increasing values of the trap's strength results in wiping out the unstable modes of the spectrum. This may be expected, as the parabolic trap makes the linearization spectrum of the system discrete (while it is continuous in the uniform space), gradually imposing a larger distance between the relevant eigenvalues, thus suppressing resonant interactions between modes that cause instabilities for such excited states [8]. Up to now, we considered the system with all the interaction (or nonlinearity) coefficients equal.…”

Section: B Numerical Resultsmentioning

confidence: 99%

Vortex-soliton complexes in coupled nonlinear Schrödinger equations with unequal dispersion coefficients

et al. 2016

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We consider a two-component, two-dimensional nonlinear Schrödinger system with unequal dispersion coefficients and self-defocusing nonlinearities, chiefly with equal strengths of the self-and cross-interactions. In this setting, a natural waveform with a nonvanishing background in one component is a vortex, which induces an effective potential well in the second component, via the nonlinear coupling of the two components. We show that the potential well may support not only the fundamental bound state, but also multi-ring excited radial state complexes for suitable ranges of values of the dispersion coefficient of the second component. We systematically explore the existence, stability, and nonlinear dynamics of these states. The complexes involving the excited radial states are weakly unstable, with a growth rate depending on the dispersion of the second component. Their evolution leads to transformation of the multi-ring complexes into stable VB solitons ones with the fundamental state in the second component. The excited states may be stabilized by a harmonic-oscillator trapping potential, as well as by unequal strengths of the self-and cross-repulsive nonlinearities.

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“…[4][5][6]. While a large volume of the early work along this vein focused on solitons and vortices, the remarkable advancement of computational resources has rendered the frontier of three-dimensional (3D) structures more accessible.…”

Section: Introductionmentioning

confidence: 99%

Robust vortex lines, vortex rings, and hopfions in three-dimensional Bose-Einstein condensates

et al. 2015

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Performing a systematic Bogoliubov-de Gennes spectral analysis, we illustrate that stationary vortex lines, vortex rings, and more exotic states, such as hopfions, are robust in three-dimensional atomic Bose-Einstein condensates, for large parameter intervals. Importantly, we find that the hopfion can be stabilized in a simple parabolic trap, without the need for trap rotation or inhomogeneous interactions. We supplement our spectral analysis by studying the dynamics of such stationary states; we find them to be robust against significant perturbations of the initial state. In the unstable regimes, we not only identify the unstable mode, such as a quadrupolar or hexapolar mode, but we also observe the corresponding instability dynamics. Furthermore, deep in the Thomas-Fermi regime, we investigate the particlelike behavior of vortex rings and hopfions.

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The Defocusing Nonlinear Schrödinger Equation

Cited by 197 publications

References 0 publications

Vector dark-antidark solitary waves in multicomponent Bose-Einstein condensates

Vector dark-antidark solitary waves in multicomponent Bose-Einstein condensates

Vortex-soliton complexes in coupled nonlinear Schrödinger equations with unequal dispersion coefficients

Robust vortex lines, vortex rings, and hopfions in three-dimensional Bose-Einstein condensates

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