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

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## Abstract

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## 1. Introduction

#### 1.1. Singular States Pulled to the Center by Attractive Fields

#### 1.1.1. Outline of the Topic

#### 1.1.2. New Results Included in the Review

#### 1.2. Singular Solitons: Previously Known Results

#### 1.2.1. Singular Solitons in Free Space

#### 1.2.2. Solitons Pinned to a Singular Potential

#### 1.2.3. New Results Included in the Review

## 2. Two-Dimensional Singular Modes in the Attractive Potential, Stabilized by the Lee–Huang–Yang (LHY) Term

#### 2.1. Analytical Approximations

#### 2.1.1. The Asymptotic Form of the Solutions at $r\to 0$ and $r\to \infty $

#### 2.1.2. The Thomas–Fermi (TF) Approximation

#### 2.2. Numerical Results for the 2D Modes Stabilized by the LHY Term

## 3. Singular Solitons in One, Two, and Three Dimensions

#### 3.1. Analytical Results

#### 3.1.1. The One-Dimensional Model with the Septimal Nonlinearity

#### 3.1.2. Physical Interpretation of the 1D Singular Soliton: Screening of a “Bare” $\delta $-Functional Potential

#### 3.1.3. The Two-Dimensional Model with the Quintic Nonlinearity

#### 3.1.4. Interpretation of the 2D Singular Soliton: Screening of a Ring-Shaped Attractive Potential

#### 3.1.5. Effects of Additional Nonlinear Terms on 1D and 2D Singular Solitons

#### 3.1.6. The 3D Model with the Cubic Nonlinearity

#### 3.1.7. Interpretation of the 3D Singular Solitons: Screening of an Attractive Spherical Potential

## 4. Numerical Results for the 1D, 2D, and 3D Singular Solitons

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

2D | two-dimensional |

3D | three-dimensional |

BEC | Bose–Einstein condensate |

GPE | Gross-Pitaevskii equation |

GS | ground state |

LHY | Lee–Huang–Yang (correction to the mean-field theory) |

MF | mean field |

NLSE | nonlinear Schrödinger equation |

TF | Thomas–Fermi (approximation) |

VK | Vakhitov–Kolokolov (stability criterion) |

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**Figure 1.**(

**a**) The profile of a (stable) numerically-found ground state (GS) and its TF counterpart, produced by Equations (30) and (38), respectively, for $\mathsf{\sigma}=0$, ${U}_{0}=10$, and $\mathsf{\mu}=-1$. The norms of the numerical and approximate solutions are ${N}_{\mathrm{num}}=41.05$, ${N}_{\mathrm{TF}}=39.68$. (

**b**) The global view of the numerical solution.

**Figure 2.**Numerically generated stable GS solutions of Equation (30) for ${U}_{0}=3;$ $\mathsf{\mu}=-0.8$; and $\mathsf{\sigma}=1$, 0, and $-1$. The respective norms are $N(\mathsf{\sigma}=1)=11.7$, $N(\mathsf{\sigma}=0)=15.41$, and $N(\mathsf{\sigma}=-1)=24.62$.

**Figure 3.**(

**a**) Radial profiles of numerically-found GSs, produced by Equation (30), and the respective interpolation profile (36), in the presence of the repulsive potential in Equation (30), with ${U}_{0}=-0.40$, at $\mathsf{\mu}=-0.8$, without and with the repulsive or attractive mean-field (MF) cubic term ($\mathsf{\sigma}=0$ and $\mathsf{\sigma}=1$ or $-1$, respectively). The corresponding values of the norm are $N\left(\mathsf{\sigma}=1\right)=0.41$, $N(\mathsf{\sigma}=0)=0.45$, and $N(\mathsf{\sigma}=-1)=0.52$. The interpolating approximation gives $N\approx 0.36$, as per Equation (37). (

**b**) Stability of the GS mode in direct simulations, in the case of $\mathsf{\sigma}=1$.

**Figure 4.**Dependencies $10\times N\left(\mathsf{\mu}\right)$ for stable GS solutions with ${U}_{0}=-0.4$, and $N\left(\mathsf{\mu}\right)$ for ${U}_{0}=$$1.5$, $3.0$, $5.0$, which correspond, respectively, to the repulsive and attractive central potential (in the former case, N is multiplied by 10, as the actual values of the norm are too small in this case). Panels (

**a**) and (

**b**) correspond to the system which does not or does include the repulsive or attractive cubic term ($\mathsf{\sigma}=0$ and $1,-1$, respectively). In (

**a**), the numerical results are juxtaposed with the TF counterparts produced by Equation (39) (except for ${U}_{0}=-0.4$, when the TF approximation is irrelevant; however, in this case the interpolating approximation, based on Equation (37), is very close to the numerically generated curve). In (

**b**), the numerical curves are compared for $\mathsf{\sigma}=1$ and $-1$. Panel (

**c**) is a zoomed-in view of the plot from (

**b**) at small values of $\left|\mathsf{\mu}\right|$, with the aim of showing relative proximity to the threshold value ${\left(\left|\mathsf{\mu}\right|\right)}_{\mathrm{thr}}=4/27\approx 0.15$ for $\mathsf{\sigma}=-1$, as predicted in the TF approximation by Equation (43) (this point is discussed in detail in the text).

**Figure 7.**(

**a**) The comparison, on the double-logarithmic scale, of the numerically found stationary solution of Equation (71) for the 3D singular soliton with $\mathsf{\mu}=-1$ (the continuous line) with profile $0.001\xb7{r}^{-1}$ relevant at small r (the long-dashed line), and the global approximation provided by Equation (77) with $C=0.001$ (the short-dashed line); see explanation in the text. The difference between the latter analytical approximation and the numerical solution is barely visible. (

**b**) The radial cross section of the 3D singular soliton, produced at $t=2$ by direct simulations of Equation (71), with the input taken from panel (

**a**). The result confirms the stability of the 3D singular soliton. In this figure, labels for $\left|\psi \right|$ and ${\mathrm{U}}_{3\mathrm{D}}$ are replaced by $\left|u\right|$.

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**MDPI and ACS Style**

Shamriz, E.; Chen, Z.; Malomed, B.A.; Sakaguchi, H.
Singular Mean-Field States: A Brief Review of Recent Results. *Condens. Matter* **2020**, *5*, 20.
https://doi.org/10.3390/condmat5010020

**AMA Style**

Shamriz E, Chen Z, Malomed BA, Sakaguchi H.
Singular Mean-Field States: A Brief Review of Recent Results. *Condensed Matter*. 2020; 5(1):20.
https://doi.org/10.3390/condmat5010020

**Chicago/Turabian Style**

Shamriz, Elad, Zhaopin Chen, Boris A. Malomed, and Hidetsugu Sakaguchi.
2020. "Singular Mean-Field States: A Brief Review of Recent Results" *Condensed Matter* 5, no. 1: 20.
https://doi.org/10.3390/condmat5010020