﻿ 二维矢量多极孤子和涡旋孤子探究 Research on Two-Dimensional Vector Multipole Solitons and Vortex Solitons

Modern Physics
Vol. 09  No. 04 ( 2019 ), Article ID: 31366 , 5 pages
10.12677/MP.2019.94021

Research on Two-Dimensional Vector Multipole Solitons and Vortex Solitons

Xianjing Lai

Department of Physics, Basic College, Zhejiang Shuren University, Hangzhou Zhejiang

Received: Jun. 20th, 2019; accepted: Jul. 15th, 2019; published: Jul. 22nd, 2019

ABSTRACT

In this paper, two-dimensional coupled nonlinear Schrödinger equations with spatial nonlinear modulation and lateral modulation are studied, and vector multipole and vortex soliton solutions are derived and analyzed. When the modulation depth is selected to be 0 and 1, the vector multipole and vortex soliton structures are obtained, respectively. The number of azimuthal lobes (the “petal" of a plurality of polarized solitons) is determined by the topological index m, and the number of layers in the multipole soliton is determined by the value of n.

Keywords:Two-Dimensional Coupled Nonlinear Schrödinger Equation, Vector Multipole Soliton, Vector Vortex Soliton

Copyright © 2019 by author(s) and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

1. 研究背景

2. 理论模型及形变约化

(1)

${\psi }_{j}\left(r,\phi ,z\right)=A\left(r\right){\Phi }_{j}\left(\phi \right)\mathrm{exp}\left(-i\kappa z\right)$ (2)

$\frac{{r}^{2}}{A}\left\{\frac{{\partial }^{2}A}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial A}{\partial r}+2\left[\kappa -R\left(r\right)\right]A-2g\left(r\right){A}^{3}\right\}={m}^{2}$ (3)

$-\frac{1}{{\varphi }_{j}}\frac{{\partial }^{2}{\varphi }_{j}}{\partial {\phi }^{2}}={m}^{2}$ (4)

${\Phi }_{j}={C}_{j}\mathrm{cos}\left(m\phi \right)+{D}_{j}\mathrm{sin}\left(m\phi \right)$ (5)

$A\left(r\right)=\rho \left(r\right)U\left[\chi \left(r\right)\right]$ 代入方程(3)，且 $U\left[\chi \left(r\right)\right]$ 满足

$-\frac{{\text{d}}^{2}U}{\text{d}{\chi }^{2}}+G\left(U\right)=\eta U$ (6)

${\rho }_{rr}+\frac{1}{r}{\rho }_{r}+\left[2\kappa -2R\left(r\right)-\frac{{m}^{2}}{{r}^{2}}\right]\rho =\frac{\eta }{{r}^{2}{\rho }^{3}}$ (7)

$\frac{G\left(U\right){\chi }_{r}^{2}}{{U}^{3}}-2g{\rho }^{2}=0$ (8)

$g\left(r\right)=G\left(U\right){r}^{-2}{\rho }^{-6}\left(r\right)/\left(2{U}^{3}\right)$$\chi \left(r\right)={\int }_{0}^{r}{\rho }^{-2}\left(s\right){s}^{-1}\text{d}s$ (9)

$\rho ={r}^{-1}\left[{c}_{1}M\left(\frac{\kappa }{2\sqrt{2\omega }},\frac{m}{2},\sqrt{2\omega }{r}^{2}\right)+{c}_{2}W\left(\frac{\kappa }{2\sqrt{2\omega }},\frac{m}{2},\sqrt{2\omega }{r}^{2}\right)\right]$ (10)

$\rho ={c}_{3}J\left(m,\sqrt{2\kappa }r\right)+{c}_{4}Y\left(m,\sqrt{2\kappa }r\right)$ (11)

$\rho =\sqrt{\frac{1}{r}\left(\alpha {\varphi }_{1}^{2}+2\beta {\varphi }_{1}{\varphi }_{2}+\gamma {\varphi }_{2}^{2}\right)}$ (12)

${\varphi }_{rr}+\left[2\kappa -2R\left(r\right)-\frac{{m}^{2}}{{r}^{2}}\right]\varphi =0$

$U\left(\chi \right)=\frac{2n\lambda }{\sqrt{-{g}_{0}}}sd\left[2n\lambda \chi \left(r\right),\frac{\sqrt{2}}{2}\right]$ (13)

3. 矢量多极孤子和涡旋孤子结构

4. 小结

Research on Two-Dimensional Vector Multipole Solitons and Vortex Solitons[J]. 现代物理, 2019, 09(04): 191-195. https://doi.org/10.12677/MP.2019.94021

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