# Dark Solitons in the Unitary Bose Gas

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## Abstract

**:**

## 1. Introduction

## 2. From Euler Equations to Nonlinear Schrödinger Equation

## 3. Nonpolynomial Schrödinger Equation

## 4. Dark Solitons in the 1D NPSE

#### 4.1. Phase

#### 4.2. Modulus

#### 4.3. Black Solitons

#### 4.4. Gray Solitons

#### 4.5. Phase

## 5. Weak vs. Strong Transverse Confinement

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Black soliton. Normalized axial density profile $\rho \left(z\right)={h}^{2}\left(z\right)={(f\left(z\right)/{f}_{\infty})}^{2}$ and transverse width profile $\sigma \left(z\right)$ vs. axial coordinate z, for three values of the parameter $\delta =\gamma |{f}_{\infty}{|}^{4/5}$, with $\gamma $ given by Equation (24) and ${f}_{\infty}$ the bulk value of the soliton wavefunction.

**Figure 2.**Gray soliton. Upper panel: Scaled axial density profile $\rho \left(\tilde{\zeta}\right)=h{\left(\tilde{\zeta}\right)}^{2}={(f\left(z\right)/{f}_{\infty})}^{2}$ vs. scaled comoving axial coordinate $\tilde{\zeta}=(z-vt)\sqrt{\delta}$. Lower panel: Scaled transverse width $\tilde{\sigma}\left(\tilde{\zeta}\right)$ vs. $\tilde{\zeta}$. Notice that $\delta =\gamma |{f}_{\infty}{|}^{4/5}$, with $\gamma $ given by Equation (24) and ${f}_{\infty}$ the bulk value of the soliton wavefunction. $\tilde{v}=v/{v}_{s}$ is the velocity rescaled by the sound velocity ${v}_{s}$.

**Figure 3.**Scaled phase $\tilde{\theta}\left(\tilde{\zeta}\right)$ of gray ($\tilde{v}>0$) and black ($\tilde{v}=0$) solitons vs. the scaled comoving axial coordinate $\tilde{\zeta}$.

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**MDPI and ACS Style**

Calzavara, M.; Salasnich, L.
Dark Solitons in the Unitary Bose Gas. *Symmetry* **2020**, *12*, 957.
https://doi.org/10.3390/sym12060957

**AMA Style**

Calzavara M, Salasnich L.
Dark Solitons in the Unitary Bose Gas. *Symmetry*. 2020; 12(6):957.
https://doi.org/10.3390/sym12060957

**Chicago/Turabian Style**

Calzavara, Martino, and Luca Salasnich.
2020. "Dark Solitons in the Unitary Bose Gas" *Symmetry* 12, no. 6: 957.
https://doi.org/10.3390/sym12060957