﻿ 二维矢量多极孤子和涡旋孤子探究

# 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

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 multi-pole 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.

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

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