Singular Mean-Field States: A Brief Review of Recent Results

Shamriz, Elad; Chen, Zhaopin; Malomed, Boris A.; Sakaguchi, Hidetsugu

doi:10.3390/condmat5010020

Cited by 22 publications

(17 citation statements)

References 70 publications

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“…In similar 1D and 3D settings, the screened charge is, respectively, diverging or vanishingly small, Q 1D ∼ ρ −1 and Q 3D ∼ ρ [50], see also a brief review of the topic in Ref. [51].…”

Section: Analytical Considerations a Asymptotic Forms Of The Solutionsmentioning

confidence: 99%

Suppression of the quasi-two-dimensional quantum collapse in the attraction field by the Lee-Huang-Yang effect

Shamriz,

Chen,

Malomed

2020

Preprint

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Quantum collapse in three and two dimensions (3D and 2D) is induced by attractive potential ∼ −r −2 . It was demonstrated that the mean-field (MF) cubic self-repulsion in the 3D bosonic gas suppresses the collapse and creates the missing ground state (GS). However, the cubic nonlinearity is not strong enough to suppress the 2D collapse. We demonstrate that the Lee-Hung-Yang (LHY) quartic term, induced by quantum fluctuations around the MF state, is sufficient for the stabilization of the 2D gas against the collapse. By means of numerical solution of the Gross-Pitaevskii equation including the LHY term, as well as with the help of analytical methods, such as expansions of the wave function at small and large distances from the center and the Thomas-Fermi approximation, we construct stable GS, with a singular density, ∼ r −4/3 but convergent integral norm. Counterintuitively, the stable GS exists even if the external potential is repulsive, with the strength falling below a certain critical value. An explanation to this finding is given. Along with the GS, singular vortex states are produced too, and their stability boundary is found analytically. Unstable vortices spontaneously transform into the stable GS, expelling the vorticity to periphery.

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Section: Analytical Considerations a Asymptotic Forms Of The Solutionsmentioning

confidence: 99%

Suppression of the quasi-two-dimensional quantum collapse in the attraction field by the Lee-Huang-Yang effect

Shamriz,

Chen,

Malomed

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…with α > 2 [11][12][13]. As demonstrated in [11], potential (2) in the GP equation for a molecular or atomic gas may be provided by attraction of a small molecule with a permanent electric dipole moment to a central charge, or attraction of a magnetically polarizable atom to electric current (such as an electron beam) piercing the plane of the 2D system (a similar but weaker singular attractive potential, ∼ −1/r, was considered under the name of a funnel potential in works [14] and [15], but it cannot create singular vortices).…”

Section: Introductionmentioning

confidence: 99%

Singular and regular vortices on top of a background pulled to the center

Chen,

Malomed

2021

Preprint

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A recent analysis has revealed singular but physically relevant localized 2D vortex states with density ∼ r −4/3 at r → 0 and a convergent total norm, which are maintained by the interplay of the potential of the attraction to the center, ∼ −r −2 , and a self-repulsive quartic nonlinearity, produced by the Lee-Huang-Yang correction to the mean-field dynamics of Bose-Einstein condensates. In optics, a similar setting, with the density singularity ∼ r −1 , is realized with the help of quintic selfdefocusing. Here we present physically relevant antidark singular-vortex states in these systems, existing on top of a flat background. Numerical solutions for them are very accurately approximated by the Thomas-Fermi wave function. Their stability exactly obeys an analytical criterion derived from analysis of small perturbations. The singular-vortex states exist as well in the case when the effective potential is weakly repulsive. It is demonstrated that the singular vortices can be excited by the input in the form of the ordinary nonsingular vortices, hence the singular modes can be created in the experiment. We also consider regular (dark) vortices maintained by the flat background, under the action of the repulsive central potential ∼ +r −2 . The dark modes with vorticities l = 0 and 1 are completely stable. In the case when the central potential is attractive, but the effective one, which includes the centrifugal term, is repulsive, and, in addition, a weak trapping potential ∼ r 2 is applied, dark vortices with l = 1 feature an intricate pattern of alternating stability and instability regions. Under the action of the instability, states with l = 1 travel along tangled trajectories, which stay in a finite area defined by the trap. The analysis is also reported for dark vortices with l = 2, which feature a complex structure of alternating intervals of stability and instability against splitting. Lastly, simple but novel flat vortices are found at the border between the anidark and dark ones.

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“…Stability of the solutions with respect of the NLS flow is then a natural question. In this respect, we also mention on the physical and modeling side the recent contributions in [26,27] that seem to have raised an interest on "singular solitons". A second relevant question concerns a detailed analysis of the evolution for the critical (p = 3) and supercritical (p > 3) nonlinearities in the n = 2 case.…”

Section: Introductionmentioning

confidence: 99%

Well posedness of the nonlinear Schrödinger equation with isolated singularities

Cacciapuoti¹,

Finco²,

Noja³

2020

Preprint

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We study the well posedness of the nonlinear Schrödinger (NLS) equation with a point interaction and power nonlinearity in dimension two and three. Behind the autonomous interest of the problem, this is a model of the evolution of so called singular solutions that are well known in the analysis of semilinear elliptic equations. We show that the Cauchy problem for the NLS considered enjoys local existence and uniqueness of strong (operator domain) solutions, and that the solutions depend continuously from initial data. In dimension two well posedness holds for any power nonlinearity and global existence is proved for powers below the cubic. In dimension three local and global well posedness are restricted to low powers.

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Singular Mean-Field States: A Brief Review of Recent Results

Cited by 22 publications

References 70 publications

Suppression of the quasi-two-dimensional quantum collapse in the attraction field by the Lee-Huang-Yang effect

Suppression of the quasi-two-dimensional quantum collapse in the attraction field by the Lee-Huang-Yang effect

Singular and regular vortices on top of a background pulled to the center

Well posedness of the nonlinear Schrödinger equation with isolated singularities

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