Collapses of Wavefunctions in Multi-Dimensional Nonlinear Schrödinger Equations under Harmonic Potential

Tsurumi, Takeya; Wadati, Miki

doi:10.1143/jpsj.66.3031

Cited by 66 publications

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“…Note that the criteria is independent of the total number of components n and any additional internal global symmetry. The known results for the single-component [26] and the multi-component [17] GPE with time-independent harmonic trap in 2 + 1 dimensions are easily reproduced from this very general result. Further, the criteria for the collapse in the system without or with the harmonic trap is also identical, except that there is no equality sign in (18) for the later case [ compare with Eq.…”

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confidence: 57%

“…However, for the multi-component case, the moment I 1 can not be identified as the total width of the wave-packet. As emphasized in our previous work [3], the moment I 1 has been used extensively in the analysis of the non-linear Schrödinger equation( NLSE) [5,6,[25][26][27], BEC [28] and in optics [29]. The dynamics of I 1 , when the system (1) is immersed in an external time-dependent harmonic trap, is the central subject of the investigation of this letter.…”

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Exact results on the dynamics of a multicomponent Bose-Einstein condensate

Ghosh

2002

Phys. Rev. A

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We study the time-evolution of the two dimensional multicomponent Bose-Einstein condensate in an external harmonic trap with arbitrary time-dependent frequency. We show analytically that the time-evolution of the total mean-square radius of the wave-packet is determined in terms of the same solvable equation as in the case of a single-component condensate. The dynamics of the total mean-square radius is also the same for the rotating as well as the non-rotating multi-component condensate. We determine the criteria for the collapse of the condensate at a finite time. Generalizing our previous work on a single-component condensate, we show explosion-implosion duality in the multi-component condensate.PACS numbers: 03.75. Fi, 05.45.Yv, 03.65.Ge The successful creation and observation of BoseEinstein condensation(BEC) in dilute alkali atoms have opened up a plethora of new possibilities to test, otherwise intractable, many-body quantum phenomenon in the laboratory [1]. The Gross-Pitaevskii equation(GPE), the mean-field description of the BEC, is successful enough in explaining most of the observed results as well as predicting new phenomenon. The methods involved in studying the GPE are mainly numerical and/or approximate: perturbative and variational. The exact and analytical results of a nonlinear equation, if known, not only act as a guide to determine the validity of different approximate and numerical methods; they also give rise to new, counter-intuitive results in some cases. Unfortunately, no exact solution of GPE is known except for in one dimension.The two dimensional GPE, like its counterparts in higher dimensions, is not exactly solvable. However, due to an underlying dynamical O(2, 1) symmetry [2], the time-evolution of certain moments related to the two dimensional GPE can be described exactly [3]. This result is valid even if the condensate is considered in a timedependent harmonic trap. This leads to the prediction of explosion-implosion duality [4] and extended parametric resonance [4,5] in the two dimensional BEC. Both of these phenomenon are universal for any non-relativistic theory having dynamical O(2, 1) symmetry [3,4,6]. Interestingly enough, apart from the two dimensional BEC, the same explosion-implosion duality can also be observed in supernova explosion and in laser induced implosion in plasma [7,8]. This shows the importance of exact methods, based on an underlying symmetry, in relating diverse areas of physics such as the BEC and the supernova explosion.The results described above are for a single-component condensate, where the spin degree's of freedom have been frozen though the use of a magnetic trap. Recently, the spinor condensate with independent spin degree's of freedom has also been created and observed in the laboratory [9]. Similarly,the two-component condensate, where two different hyperfine states of the same atomic species are condensed simultaneously, has also been experimentally realized [10]. The spinor condensate [11][12][13][14] and the two-component condensate [15] ...

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Exact results on the dynamics of a multicomponent Bose-Einstein condensate

Ghosh

2002

Phys. Rev. A

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“…It is also possible to argue that the collapse does not occur in this system using the more conventional language of momenta [7,26]. To this end let us define the mean squared width of the wave packet…”

Section: Strongly Nonlocal Limitmentioning

confidence: 99%

“…In this framework, the interaction between the constitutive bosons inside the condensate is defined in terms of the ground state scattering length a. When a > 0 the interaction between the particles in the condensate is repulsive which reflects into a positive self-interaction term in the GP equation, whereas for a < 0, the interaction is attractive and selfinteraction leads to collapse in dimension higher or equal than two [5][6][7]. It is thought that the practical implication is that, in this case, the realization of the condensate is limited by a critical number of particles [8] above which the condensate is unstable and destroyed by the collapse phenomenon [8,[5][6][7].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of quasicollapse in nonlinear Schrödinger systems with nonlocal interactions

2000

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We study the effect of nonlocality on the collapse properties of a self-focusing Nonlinear Schrödinger system related to Bose-Einstein condensation problems. Using a combination of moment techniques, time dependent variational methods and numerical simulations we present evidences in support of the hypothesis that nonlocal attractively interacting condensates cannot collapse when the dominant interaction term is due to finite range interactions. Instead there apppear oscillations of the wave packet with a localized component whose size is of the order of the range of interactions. We discuss the implications of the results to collapse phenomena in negative scattering length Bose-Einstein condensates.

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“…When γ > 0, it can be thought of as modeling inhomogeneities in the medium. The nonlinearity enters due to the effect of changes in the field intensity on the wave propagation characteristics of the medium and the nonlinear weight can be looked as the proportional to the electron density [18,33,38].…”

Section: Introductionmentioning

confidence: 99%

Remarks on the inhomogeneous fractional nonlinear Schrödinger equation

Saanouni

2016

Journal of Mathematical Physics

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Abstract. Using a sharp Gagliardo-Nirenberg type inequality, well-posedness issues of the initial value problem for a fractional inhomogeneous Schrödinger equation are investigated. Consider the initial value problem for an inhomogeneous nonlinear Schrödinger equation Contentswhich models various physical contexts in the description of nonlinear waves such as propagation of a laser beam and plasma waves. For example, when γ = 0, it arises in nonlinear optics, plasma physics and fluid mechanics [2,3]. When γ > 0, it can be thought of as Date: February 16, 2016. 1991 Mathematics Subject Classification. 35Q55.

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Collapses of Wavefunctions in Multi-Dimensional Nonlinear Schrödinger Equations under Harmonic Potential

Cited by 66 publications

References 7 publications

Exact results on the dynamics of a multicomponent Bose-Einstein condensate

Exact results on the dynamics of a multicomponent Bose-Einstein condensate

Dynamics of quasicollapse in nonlinear Schrödinger systems with nonlocal interactions

Remarks on the inhomogeneous fractional nonlinear Schrödinger equation

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