Stationary solutions of the one-dimensional nonlinear Schrödinger equation. II. Case of attractive nonlinearity

Carr, Lincoln D.; Clark, Charles W.; Reinhardt, William P.

doi:10.1103/physreva.62.063611

Cited by 262 publications

(222 citation statements)

References 22 publications

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“…We conclude our stability analysis by considering the short time behavior of the stationary states under the action of an initial finite deformation. A similar analysis has been considered in [21] to check the stability of the solutions found in [10]. The authors of [21] studied the evolution of stationary states initially perturbed with a stochastic noise.…”

Section: Stability Of Stationary Solutionsmentioning

confidence: 99%

“…In particular, the appearance of self trapping stationary states in the dimer case, which mimics a double-well system, has been widely investigated in connection with the evolution of wave packets [12,13,14]. Recently, a set of stationary solutions without linear counterpart has been discovered also in the continuous case, namely the exactly solvable 1-D GPE with periodic boundary conditions and zero external potential [10]. These states break the rotational invariance of the associated linear problem.…”

Section: Introductionmentioning

confidence: 99%

“…stationary solutions of the GPE which can be obtained as a deformation of eigenstates of the corresponding linear Schrödinger equation [9,10]. However, the GPE may also admit stationary solutions without linear counterpart.…”

Section: Introductionmentioning

confidence: 99%

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States without a linear counterpart in Bose-Einstein condensates

D’Agosta

Presilla

2002

Phys. Rev. A

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We show the existence of stationary solutions of a 1-D Gross-Pitaevskii equation in presence of a multi-well external potential that do not reduce to any of the eigenfunctions of the associated Schrödinger problem. These solutions, which in the limit of strong nonlinearity have the form of chains of dark or bright solitons located near the extrema of the potential, represent macroscopically excited states of a Bose-Einstein condensate and are in principle experimentally observable.

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Section: Stability Of Stationary Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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States without a linear counterpart in Bose-Einstein condensates

D’Agosta

Presilla

2002

Phys. Rev. A

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“…Static and dynamical properties of Bose-Einstein condensates have been extensively and successfully explored in the community by employing the mean-field Gross-Pitaevskii theory [11,12], for reviews see [8][9][10] and for individual applications to condensates and mixtures [13][14][15][16][17][18][19] and [20][21][22][23][24][25], respectively. Gross-Pitaevskii theory is an excellent theory for weaklyinteracting bosons whenever a single macroscopic one-particle wavefunction is sufficient to describe the reality.…”

Section: Introductionmentioning

confidence: 99%

Multiconfigurational time-dependent Hartree method for bosons: Many-body dynamics of bosonic systems

2008

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The evolution of Bose-Einstein condensates is amply described by the time-dependent GrossPitaevskii mean-field theory which assumes all bosons to reside in a single time-dependent oneparticle state throughout the propagation process. In this work, we go beyond mean-field and develop an essentially-exact many-body theory for the propagation of the time-dependent Schrödinger equation of N interacting identical bosons. In our theory, the time-dependent many-boson wavefunction is written as a sum of permanents assembled from orthogonal one-particle functions, or orbitals, where both the expansion coefficients and the permanents (orbitals) themselves are timedependent and fully determined according to a standard time-dependent variational principle. By employing either the usual Lagrangian formulation or the Dirac-Frenkel variational principle we arrive at two sets of coupled equations-of-motion, one for the orbitals and one for the expansion coefficients. The first set comprises of first-order differential equations in time and non-linear integro-differential equations in position space, whereas the second set consists of first-order differential equations with time-dependent coefficients. We call our theory multi-configurational timedependent Hartree for bosons, or MCTDHB(M ), where M specifies the number of time-dependent orbitals used to construct the permanents. Numerical implementation of the theory is reported and illustrative numerical examples of many-body dynamics of trapped Bose-Einstein condensates are provided and discussed.

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“…Consider condensed atoms confined in a toroidal trap of radius R and thickness r, where r ≪ R so that lateral motion is negligible and the system is essentially one-dimensional [17]. The dynamics of the BEC is described by the dimensionless nonlinear Gross-Pitaveskii (GP) equation,…”

Section: Physical Model: Kicked Bec On a Ringmentioning

confidence: 99%

Transition to instability in a periodically kicked Bose-Einstein condensate on a ring

et al. 2006

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A periodically kicked ring of a Bose-Einstein condensate is considered as a nonlinear generalization of the quantum kicked rotor, where the nonlinearity stems from the mean field interactions between the condensed atoms. For weak interactions, periodic motion (anti-resonance) becomes quasiperiodic (quantum beating) but remains stable. There exists a critical strength of interactions beyond which quasiperiodic motion becomes chaotic, resulting in an instability of the condensate manifested by exponential growth in the number of noncondensed atoms. In the stable regime, the system remains predominantly in the two lowest energy states and may be mapped onto a spin model, from which we obtain an analytic expression for the beat frequency and discuss the route to instability. We numerically explore parameter regime for the occurrence of instability and reveal the characteristic density profile for both condensed and noncondensed atoms. The Arnold diffusion to higher energy levels is found to be responsible for the transition to instability. Similar behavior is observed for dynamically localized states (essentially quasiperiodic motions), where stability remains for weak interactions but is destroyed by strong interactions.

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Stationary solutions of the one-dimensional nonlinear Schrödinger equation. II. Case of attractive nonlinearity

Cited by 262 publications

References 22 publications

States without a linear counterpart in Bose-Einstein condensates

States without a linear counterpart in Bose-Einstein condensates

Multiconfigurational time-dependent Hartree method for bosons: Many-body dynamics of bosonic systems

Transition to instability in a periodically kicked Bose-Einstein condensate on a ring

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