# Acquiring the Symplectic Operator Based on Pure Mathematical Derivation Then Verifying It in the Intrinsic Problem of Nanodevices

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Calculate Symplectic Operators

_{1}is

_{1}is shown in Figure 3.

#### 2.2. Comparison with Different Symplectic Operators

## 3. Application in the Intrinsic Problem of Nanodevices

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**In the second-order, different symplectic operators are represented in the image with d

_{1}. It can be found that different symplectic operators change greatly with the value of d

_{1}, thus affecting the whole algorithm.

**Figure 2.**The second-order symplectic operator error function image, we can find from the image that the error function fluctuates with the value of d

_{1}and there is a minimum in a certain range.

**Figure 3.**The variation of the symplectic operators with the d

_{1}in the third-order symplectic algorithm.

**Figure 4.**Comparing the symplectic algorithms of different orders, it can be clearly seen that the fourth-order symplectic algorithm has the best dispersion characteristics.

**Figure 5.**Comparing the different levels of the symplectic algorithm, it can be clearly seen that the higher the number of stages, the better the dispersion characteristics of the symplectic algorithm.

**Figure 6.**By Comparing the symplectic algorithm of the same order number obtained by the current research with the symplectic algorithm obtained by Ruth [4], we can see that our method is significantly better than the traditional method.

**Table 1.**Value of symplectic operator propagation coefficient. SFDTD, Symplectic Finite-Difference Time-Domain.

(c,d) | c | d |
---|---|---|

SFDTD (2,4) | ${c}_{1}=0.6891,$ ${c}_{2}=0.3207$ | ${d}_{1}=-2.70309412,$ ${d}_{2}=-0.53652708,$ ${d}_{3}=2.37893931,$ ${d}_{4}=1.86068189$ |

SFDTD (4,2) | ${c}_{1}=0.3163393038,$ ${c}_{2}=-0.081630997,$ ${c}_{3}=0.53047572287179,$ ${c}_{4}=0.20367456$ | ${d}_{1}=0.7432,$ ${d}_{2}=0.3021$ |

SFDTD (4,3) | ${c}_{1}=0.81431,$ ${c}_{2}=-0.31431,$ ${c}_{3}=-0.31431,$ ${c}_{4}=0.81431$ | ${d}_{1}=-0.093769,$ ${d}_{2}=1.18653,$ ${d}_{3}=-0.093769$ |

SFDTD (4,4) | ${c}_{1}=0.2581,$ ${c}_{2}=1.3821,$ ${c}_{3}=-0.2403,$ ${c}_{4}=-0.5317$ | ${d}_{1}=0.4311,$ ${d}_{2}=-0.0722,$ ${d}_{3}=0.1411,$ ${d}_{4}=0.5013$ |

Ruth (4,4) | ${c}_{1}=0.6756,$ ${c}_{2}=-0.1756,$ ${c}_{3}=-0.1756,$ ${c}_{4}=0.6756$ | ${d}_{1}=1.3512,$ ${d}_{2}=-1.7024,$ ${d}_{3}=1.3512,$ ${d}_{4}=0$ |

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**MDPI and ACS Style**

Nie, H.; Gui, R.; Chen, T.
Acquiring the Symplectic Operator Based on Pure Mathematical Derivation Then Verifying It in the Intrinsic Problem of Nanodevices. *Symmetry* **2019**, *11*, 1383.
https://doi.org/10.3390/sym11111383

**AMA Style**

Nie H, Gui R, Chen T.
Acquiring the Symplectic Operator Based on Pure Mathematical Derivation Then Verifying It in the Intrinsic Problem of Nanodevices. *Symmetry*. 2019; 11(11):1383.
https://doi.org/10.3390/sym11111383

**Chicago/Turabian Style**

Nie, Han, Renzhou Gui, and Tongjie Chen.
2019. "Acquiring the Symplectic Operator Based on Pure Mathematical Derivation Then Verifying It in the Intrinsic Problem of Nanodevices" *Symmetry* 11, no. 11: 1383.
https://doi.org/10.3390/sym11111383