# Unique Features of Nonlocally Nonlinear Systems with Oscillatory Responses

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Nonlocality-Controllable Kerr Nonlinearities

## 3. Unique Modulational Instability

#### 3.1. Short-Term Evolution of MI

#### 3.2. Long-Term Evolution of MI

## 4. Characteristics of Solitons

#### 4.1. Fundamental Solitons

#### 4.2. In-Phase and Out-of-Phase Bound-State Solitons

#### 4.3. Multi-Peak Solitons

## 5. Perturbation-Iteration Method

## 6. Optical Beams in NLC with Negative Dielectric Anisotropy

#### 6.1. Evolution Equation for Optical Beams in NLC

#### 6.2. Optical Nonlinearities of NLC with Negative Dielectric Anisotropy

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Beam widths at $z=1$ as the function of ${\sigma}_{0}$ for different input powers. (

**a**) $D=1$, (

**b**) $D=2$. Solid color curves represent the numerical simulation results, agreeing well with the variational results denoted by dashed color curves. For comparison, the output beam width at $z=1$ in the linear case is also plotted by the horizontal solid straight line (After Ref. [28]).

**Figure 3.**Evolutions of optical beams for both $s=-1$ (${\mathbf{a}}_{\mathbf{1}}$,${\mathbf{a}}_{\mathbf{3}}$) and $s=1$ (${\mathbf{b}}_{\mathbf{1}}$,${\mathbf{b}}_{\mathbf{3}}$). ${\sigma}_{0}=0.45$ in (${\mathbf{a}}_{\mathbf{1}}$,${\mathbf{b}}_{\mathbf{1}}$), whereas ${\sigma}_{0}=1.2$ in (${\mathbf{a}}_{\mathbf{3}}$,${\mathbf{b}}_{\mathbf{3}}$). The linear evolutions are displayed in (${\mathbf{a}}_{\mathbf{2}}$,${\mathbf{b}}_{\mathbf{2}}$) for comparison (After Ref. [27]).

**Figure 7.**Evolution of stable solution when ${w}_{m}=1$ at different modulation frequencies: (

**a**) ${k}_{x}=1.6$; (

**b**) ${k}_{x}=1.4$ (After Ref. [32]).

**Figure 8.**Critical power ${P}_{c}$ and the soliton propagation constant $\lambda $ versus the degree of nonlocality $\sigma $. The inset shows ${P}_{c}$ versus $\lambda $. All plots are for $D=1$. The obtained solitons are stable (After Ref. [27]).

**Figure 9.**Profiles of in-phase bound-state solitons (solid red lines) and their induced NRI (dashed black lines) for the case that $s=1$. (

**a**) ${\sigma}_{0}=0.472$, (

**b**) ${\sigma}_{0}=0.700$, (

**c**) ${\sigma}_{0}=0.773$ and (

**d**) ${\sigma}_{0}=0.778$. The obtained solitons are stable (After Ref. [29]).

**Figure 10.**Profile of out-of-phase bound-state solitons (solid red lines) and the induced NRI (dashed black lines) for the case that $s=1$. (

**a**) ${\sigma}_{0}=0.453$, (

**b**) ${\sigma}_{0}=0.731$, (

**c**) ${\sigma}_{0}=0.777$ and (

**d**) ${\sigma}_{0}=0.778$. The obtained solitons are stable (After Ref. [29]).

**Figure 11.**(

**a**) Power P and propagation constant $\lambda $ vs. the degree of nonlocality $\sigma $; (

**b**) power P vs. propagation constant $\lambda $ of the in-phase and out-of-phase bound-state solitons for the case that $s=1$. The obtained solitons are stable (After Ref. [29]).

**Figure 12.**Profiles of the numerical multi–peak solitons $u\left(x\right)$ (the solid red curves) for $s=-1$ and ${w}_{m}=5$. (

**a**–

**d**) for $n=1,4,5,6$ and ${\sigma}_{0}=1.13,1.43,3.52,1.08$, respectively. The variational results (the dashed black curves) with the same parameters are also given for comparison. Soliton in (

**b**) is stable, while the ones in the other three figures are unstable.

**Figure 13.**Profiles of the numerical multi–peak solitons $u\left(x\right)$ (the solid red curves) for $s=1$ and ${w}_{M}=10$. (

**a**–

**d**) for $n=2,3,5,6$ and ${\sigma}_{0}=0.74,0.48,0.75,0.51$, respectively. The corresponding numerical results are denoted in the same manner as in Figure 5. Soliton in (

**b**) is stable, while the ones in the other three figures are unstable.

**Figure 14.**Dependencies of the power ${P}_{c}$ on the degree of nonlocality ${\sigma}_{0}$ (

**a**,

**c**) and the propagation constant b on the degree of nonlocality ${\sigma}_{0}$ (

**b**,

**d**) for the multi–peak solitons. (

**a**,

**b**) and (

**c**,

**d**) for $s=-1$ and $s=1$, respectively.)

**Figure 15.**X-Z cross-section of the planar cell of the NLC with negative dielectric anisotropy, and the cell can be considered invariant along Y. Two pieces of glasses sandwiches NLC. Polymer coatings provide molecules anchoring ${\theta |}_{X=L/2}{=\theta |}_{X=-L/2}=\pi /2$ at the boundaries, and ITO films allow for the application of low frequency bias. An optical beam propagating inside the cell along Z induces an index perturbation.

**Figure 16.**Dependence of ${W}_{mL}/L$ on bias voltage ${V}_{rf}$ for different NLC samples. The values of ${\u03f5}_{a}^{rf}$ are $-5.9$, $-5.3$ and $-3.8$ for KY1-008, KY19-008 and KY6-008-type NLCs, respectively. $K=1\times {10}^{-11}N$ for all materials.

$\mathit{n}=0$ | $\mathit{n}=1$ | $\mathit{n}=2$ | $\mathit{n}\ge 3$ | ||
---|---|---|---|---|---|

$s=-1$ | variational | $(1.05,+\infty )$ | $(1.06,+\infty )$ | $(1.06,+\infty )$ | $(1.06,+\infty )$ |

numerical | $(1.05,+\infty )$ | $(1.05,+\infty )$ | $(1.05,+\infty )$ | $(1.05,+\infty )$ | |

$s=+1$ | variational | $(0,0.77)$ | $(0.38,0.79)$ | $(0.39,0.79)$ | $(0.39,0.79)$ |

numerical | $(0.05,0.78)$ | $(0.38,0.78)$ | $(0.39,0.78)$ | $(0.40,0.78)$ |

_{0}∈ (σ

_{c}, +∞) for s = −1, and [0, σ

_{c}] for s = 1 (for example, ${{\sigma}_{c}|}_{z=1}$ = 0.82.)

$\mathit{n}=0$ | $\mathit{n}=1$ | $\mathit{n}=2$ | $\mathit{n}=3$ | $\mathit{n}=4$ | $\mathit{n}\ge 5$ | |
---|---|---|---|---|---|---|

$s=-1$ | $(1.05,+\infty )$ | $(1.41,+\infty )$ | $(1.10,+\infty )$ | $(1.10,+\infty )$ | $(1.10,1.61)$ | no |

$s=1$ | $(0.05,0.78)$ | $(0.38,0.78)$ | $(0.42,0.45)$ | $(0.48,0.49)$ | no | no |

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**MDPI and ACS Style**

Liang, G.; Liu, J.; Hu, W.; Guo, Q.
Unique Features of Nonlocally Nonlinear Systems with Oscillatory Responses. *Appl. Sci.* **2022**, *12*, 2386.
https://doi.org/10.3390/app12052386

**AMA Style**

Liang G, Liu J, Hu W, Guo Q.
Unique Features of Nonlocally Nonlinear Systems with Oscillatory Responses. *Applied Sciences*. 2022; 12(5):2386.
https://doi.org/10.3390/app12052386

**Chicago/Turabian Style**

Liang, Guo, Jinlong Liu, Wei Hu, and Qi Guo.
2022. "Unique Features of Nonlocally Nonlinear Systems with Oscillatory Responses" *Applied Sciences* 12, no. 5: 2386.
https://doi.org/10.3390/app12052386