# Small-Amplitude Nonlinear Modes under the Combined Effect of the Parabolic Potential, Nonlocality and PT Symmetry

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Models

## 3. Bifurcations of Nonlinear Modes

## 4. Linear Stability of Nonlinear Modes

#### 4.1. Statement of the Problem

#### 4.2. Linear Stability of Small-Amplitude Modes

## 5. Results for the Real Parabolic Potential

## 6. Results for the $\mathcal{PT}$-Symmetric Parabolic Potential

## 7. Nonlinear Modes of Finite Amplitude

## 8. Conclusions

- A set of continuous families of nonlinear modes exists in the nonlocal $\mathcal{PT}$-symmetric nonlinear Schrödinger equation. The spectrum of the corresponding linear eigenvalue problem is available in the analytical form, and thus the stability problem for the small-amplitude nonlinear modes can be reduced to the searching real roots of certain analytical expressions.
- In the conservative case, small-amplitude nonlinear modes of arbitrary order become stable for sufficiently strong nonlocality, provided that the kernel is a sufficiently smooth function in the vicinity of the origin. If the kernel feature a singularity at $x=0$, no stabilization of higher-order nonlinear modes is observed.
- Both the degree of nonlocality and the strength of the $\mathcal{PT}$ symmetry can be used to manage stability of the small-amplitude modes.
- The above stability conclusions remain valid for nonlinear modes of finite amplitude.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

$\mathcal{PT}$ | Parity-Time |

BEC | Bose-Einstein condensate |

## Appendix A. Examples of Expressions for ${D}_{n,k}$

## Appendix B. Definition and Some Properties of the Fourier Transform

## Appendix C. Some Properties of the Hermite—Gauss Eigenfunctions

## Appendix D. Asymptotics in the Strongly Nonlocal Limit

**Asymptotics for ${b}_{n}^{\left(2\right)}$.**

**Asymptotics for ${\left({M}_{n,k}\right)}_{1,1}$.**

**Asymptotics for ${\left({M}_{n,k}\right)}_{2,1}$.**

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**Figure 1.**${L}^{2}$-normalized linear modes (6) for $n=0,1,2$ and $\alpha =0,1/2,1$. Red, magenta and black lines correspond to Re ${\tilde{w}}_{n}$, Im ${\tilde{w}}_{n}$, and $|{\tilde{w}}_{n}|$, respectively. For all panels the axes boxes correspond to the interval $[-5,5]$ (horizontal axes) and $[-0.8,0.8]$ (vertical axes).

**Figure 2.**Linear limit stability diagrams for $n=2$ and $n=3$ in the plane $(\alpha ,\lambda )$. Darker domains correspond to instability. Labels “$k=0$”, “$k=1$”, etc. at the unstable domains show the number of the double eigenvalue ${\Omega}_{n,k}$ responsible for the instability.

**Figure 3.**The families of nonlinear modes with $n=0,1,2,3$ for different α and λ (the Gaussian kernel). Stable modes correspond to the bold fragments of the curves.

n | k | $\mathbf{Sign}{\left.{\mathit{D}}_{\mathit{n},\mathit{k}}\left(\mathit{\lambda}\right)\right|}_{\mathit{\lambda}=\mathbf{0}}$ | Zeros of ${\mathit{D}}_{\mathit{n},\mathit{k}}\left(\mathit{\lambda}\right)$ | Interval of Stability in λ |
---|---|---|---|---|

1 | 0 | + | no zeros | $\lambda \in [0,\infty )$ |

2 | 0 | − | 1.82 | $\lambda \in (1.82,\infty )$ |

1 | + | no zeros | ||

3 | 0 | − | 2.36 | $\lambda \in (2.36,\infty )$ |

1 | − | 2.04 | ||

2 | + | no zeros | ||

4 | 0 | + | 2.61 2.85 | $\lambda \in (2.85,\infty )$ |

1 | − | 2.64 | ||

2 | − | 2.26 | ||

3 | + | no zeros | ||

5 | 0 | + | 3.23 3.25 | $\lambda \in (3.12,3.23)\cup (3.25,\infty )$ |

1 | − | 3.12 | ||

2 | − | 2.90 | ||

3 | − | 2.45 | ||

4 | + | no zeros | ||

6 | 0 | + | 3.566 3.568 | $\lambda \in (3.37,3.45)\cup (3.51,3.566)\cup (3.568,\infty )$ |

1 | + | 3.45 3.51 | ||

2 | − | 3.37 | ||

3 | − | 3.13 | ||

4 | − | 2.64 | ||

5 | + | no zeros |

n | k | $\mathbf{Sign}{\left.{\mathit{D}}_{\mathit{n},\mathit{k}}\left(\mathit{\lambda}\right)\right|}_{\mathit{\lambda}=\mathbf{0}}$ | Zeros of ${\mathit{D}}_{\mathit{n},\mathit{k}}\left(\mathit{\lambda}\right)$ | Interval of Stability in λ |
---|---|---|---|---|

1 | 0 | + | no zeros | $\lambda \in [0,\infty )$ |

2 | 0 | − | 1.31 | $\lambda \in (1.31,\infty )$ |

1 | + | no zeros | ||

3 | 0 | − | no zeros | always unstable |

1 | − | 1.54 | ||

2 | + | no zeros | ||

4 | 0 | + | 7.25 | always unstable |

1 | − | no zeros | ||

2 | − | 1.70 | ||

3 | + | no zeros | ||

5 | 0 | + | no zeros | always unstable |

1 | − | no zeros | ||

2 | − | no zeros | ||

3 | − | 1.84 | ||

4 | + | no zeros | ||

6 | 0 | + | no zeros | always unstable |

1 | + | no zeros | ||

2 | − | no zeros | ||

3 | − | no zeros | ||

4 | − | 1.97 | ||

5 | + | no zeros |

n | k | $\mathbf{Sign}{\left.{\mathit{D}}_{\mathit{n},\mathit{k}}\left(\mathit{\lambda}\right)\right|}_{\mathit{\lambda}=\mathbf{0}}$ | Zeros of ${\mathit{D}}_{\mathit{n},\mathit{k}}\left(\mathit{\lambda}\right)$ | Interval of Stability in λ |
---|---|---|---|---|

1 | 0 | + | no zeros | $\lambda \in [0,\infty )$ |

2 | 0 | − | 1.52 | $\lambda \in (1.52,\infty )$ |

1 | + | no zeros | ||

3 | 0 | − | 2.70 | $\lambda \in (2.70,\infty )$ |

1 | − | 1.66 | ||

2 | + | no zeros | ||

4 | 0 | + | 2.77 3.45 | $\lambda \in (3.45,\infty )$ |

1 | − | 2.96 | ||

2 | − | 1.77 | ||

3 | + | no zeros | ||

5 | 0 | + | 3.88 3.99 | $\lambda \in (3.74,3.88)\cup (3.99,\infty )$ |

1 | − | 3.74 | ||

2 | − | 3.21 | ||

3 | − | 1.87 | ||

4 | + | no zeros | ||

6 | 0 | + | 4.39 4.42 | $\lambda \in (4.28,4.39)\cup (4.42,\infty )$ |

1 | + | 3.98 4.28 | ||

2 | − | 4.00 | ||

3 | − | 3.44 | ||

4 | − | 1.97 | ||

5 | + | no zeros |

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**MDPI and ACS Style**

Zezyulin, D.A.; Konotop, V.V.
Small-Amplitude Nonlinear Modes under the Combined Effect of the Parabolic Potential, Nonlocality and PT Symmetry. *Symmetry* **2016**, *8*, 72.
https://doi.org/10.3390/sym8080072

**AMA Style**

Zezyulin DA, Konotop VV.
Small-Amplitude Nonlinear Modes under the Combined Effect of the Parabolic Potential, Nonlocality and PT Symmetry. *Symmetry*. 2016; 8(8):72.
https://doi.org/10.3390/sym8080072

**Chicago/Turabian Style**

Zezyulin, Dmitry A., and Vladimir V. Konotop.
2016. "Small-Amplitude Nonlinear Modes under the Combined Effect of the Parabolic Potential, Nonlocality and PT Symmetry" *Symmetry* 8, no. 8: 72.
https://doi.org/10.3390/sym8080072