# Exact Solutions for Solitary Waves in a Bose-Einstein Condensate under the Action of a Four-Color Optical Lattice

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## Abstract

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## 1. Introduction

## 2. Exact Analytical Model for Obtaining the Solitary Excitations under the Novel FOL Trap

## 3. The Parameter Domain and Shape of the Tunable FOL

## 4. Density Patterns Supported by the Engineered FOL

## 5. Dynamical Stability and Structural Stability of the Condensate

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**Curves and points for $l=0.84$ where the potential is not a FOL, but a TOL or a BOL. ‘×’ signifies no potential for ${b}_{1}={b}_{2}=0$. All other points in the $\left({b}_{1},{b}_{2}\right)$ plane correspond to FOLs.

**Figure 2.**The variation of the FOL potential following the change of the tuning parameters for $l=0.84$: (

**a**) for fixed ${b}_{2}=2$, ${b}_{1}$ varies from $-4$ to $+4$; (

**b**) for fixed ${b}_{1}=2$, ${b}_{2}$ varies from $-4$ to $+4$. Here and in the figures following below, the results are displayed in interval $-14<z<+14$.

**Figure 3.**Condensate density patterns for ${b}_{1}>0$ and ${b}_{2}>0$: (

**a**) ${b}_{1}=1$; ${b}_{2}=2$, (

**b**) ${b}_{1}=1$; ${b}_{2}=3.5$, (

**c**) ${b}_{1}=2$; ${b}_{2}=1$, and (

**d**) ${b}_{1}=3.5$; ${b}_{2}=1$. Each plot of (

**a**–

**d**) has two panels: the upper panel shows the contour plot of the density and the lower panel consists of a 2D plot of the density combined with the corresponding potential profile.

**Figure 4.**Condensate density patterns for ${b}_{1}<0$ or ${b}_{2}<0$ or both: (

**a**) ${b}_{1}=-1$; ${b}_{2}=1$, (

**b**) ${b}_{1}=-3$; ${b}_{2}=1$, (

**c**) ${b}_{1}=1$; ${b}_{2}=-3$, and (

**d**) ${b}_{1}=-1$; ${b}_{2}=-3$. Each plot of (

**a**–

**d**) has two panels: the upper panel shows the contour plot of the density and the lower panel consists of a 2D plot of the density combined with the corresponding potential profile.

**Figure 5.**Condensate densities are depicted by filled plots at times (

**a**) $t=0$, (

**b**) $t=$ 10 ms, and (

**c**) $t=$ 20 ms, along with the potential energy profile (solid-line curve) for ${b}_{1}=2$ and ${b}_{2}=1$. Initial density (dotted curve) is merged with the densities in (

**b**,

**c**) for reference.

**Figure 6.**The numerical stability analysis of one of the obtained solutions with ${b}_{1}=2$ and ${b}_{2}=1$. The condensate density is depicted by the dotted line, and the trap profile, $V\left(z\right)$ (not in scale), is superimposed on it (the solid line). The deviation of the noisy data from their noise-free counterparts is shown for both kinds of the analyses: the dynamical stability, ${D}_{W}$ (the upper curve, composed of symbols *), and the structural stability, ${D}_{P}$ (the lower curve, composed of symbols ⨁).

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**MDPI and ACS Style**

Halder, B.; Ghosh, S.; Basu, P.; Bera, J.; Malomed, B.; Roy, U.
Exact Solutions for Solitary Waves in a Bose-Einstein Condensate under the Action of a Four-Color Optical Lattice. *Symmetry* **2022**, *14*, 49.
https://doi.org/10.3390/sym14010049

**AMA Style**

Halder B, Ghosh S, Basu P, Bera J, Malomed B, Roy U.
Exact Solutions for Solitary Waves in a Bose-Einstein Condensate under the Action of a Four-Color Optical Lattice. *Symmetry*. 2022; 14(1):49.
https://doi.org/10.3390/sym14010049

**Chicago/Turabian Style**

Halder, Barun, Suranjana Ghosh, Pradosh Basu, Jayanta Bera, Boris Malomed, and Utpal Roy.
2022. "Exact Solutions for Solitary Waves in a Bose-Einstein Condensate under the Action of a Four-Color Optical Lattice" *Symmetry* 14, no. 1: 49.
https://doi.org/10.3390/sym14010049