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In this paper, we deeply investigate the phase evolution and the underlying topological vector potential in the nonlinear interference of solitons. Based on the double-soliton solution of 1D nonlinear Schrödinger equation, we find that the density zeros of wave function generally exist in the extended complex space, each density zero corresponds to the vector potential produced by Dirac magnetic monopole. The vector potential field is composed of periodically distributed Dirac magnetic monopole pairs with opposite magnetic charges. By observing the motion of magnetic monopoles, we can conveniently understand the phase evolution characteristics during the interference process. In particular, we find that the collision of a pair of magnetic monopoles with opposite charge on the real axis corresponds exactly to the
$ \pm\pi $ jump of the wave function phase at nodes. For comparison, we also discuss Dirac magnetic monopoles and vector potential field in linear wave packet interference case. The results show that the Dirac magnetic monopole potential widely exists in the interference phenomena of wave fields, and the distribution of magnetic monopoles in the extended complex space can be used to distinguish the topological properties behind the linear and nonlinear interference process.-
Keywords:
- monopole /
- nonlinear /
- interference /
- soliton
[1] Dirac P A M 1931 Proc. R. Soc. Lond. A 133 60
Google Scholar
[2] Milton K A 2006 Rep. Prog. Phys. 69 1637
Google Scholar
[3] Yang C N 1970 Phys. Rev. D 1 2360
Google Scholar
[4] Wu T T, Yang C N 1995 Phys. Rev. D 12 3845
Google Scholar
[5] Berry M V 1980 Eur. J. Phys. 1 240
Google Scholar
[6] Aharonov Y, Bohm D 1959 Phys. Rev. 115 485
Google Scholar
[7] Berry M V 1984 Proc. R. Soc. Lond. A 392 45
Google Scholar
[8] Hooft G 1974 Nucl. Phys. B 79 276
Google Scholar
[9] Castelnovo C, Moessner R, Sondhi S L 2008 Nature 451 42
Google Scholar
[10] Milde P, Köhler D, Seidel J, Eng L M, Bauer A, Chacon A, Kindervater J, Mühlbauer S, Pfleiderer C, Buhrandt S, Schütte C, Rosch A 2013 Science 340 1076
Google Scholar
[11] Ray M W, Ruokokoski E, Kandel S, Möttönen M, Hall D S 2014 Nature 505 657
Google Scholar
[12] Xiao D, Chang M C, Niu Q 2010 Rev. Mod. Phys. 82 1959
Google Scholar
[13] Zhao L C, Qin Y H, Lee C, Liu J 2021 Phys. Rev. E 10 3
Google Scholar
[14] Muga J G, Ruschhaupt A, Campo A 2009 Time in Quantum Mechanics (Vol. 2) (Berlin, Heidelberg: Springer Berlin Heidelberg) p305
[15] Zabusky N J, Kruskal M D 1965 Phys. Rev. Lett. 15 240
Google Scholar
[16] Barenblatt G I 1996 Scaling, Self-similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics (Cambridge: Cambridge University Press)
[17] Karpman V I 1975 Non-Linear Waves in Dispersive Media (New York: Pergamon Press)
[18] Agrawal G 2006 Nonlinear Fiber Optics (Cambridge: Academic Press)
[19] Wu B, Liu J, Niu Q 2002 Phys. Rev. Lett. 88 034101
Google Scholar
[20] Rebbi C, Soliani G 1984 Solitons and Particles (Singapore: World Scientific Publishing)
[21] Nguyen J H V, Dyke P, Luo D, Malomed B A, Hulet R G 2014 Nat. Phys. 10 918
Google Scholar
[22] Zakharov V E, Shabat A B 1973 Sov. Phys. JETP 37 823
[23] Zhao L C, Ling L, Yang Z Y, Liu J 2016 Nonlinear Dyn. 83 659
Google Scholar
[24] Yang C N, Lee T D 1952 Phys. Rev. 87 404
Google Scholar
[25] Zhao L C, Meng L Z, Qin Y H, Yang Z Y, Liu J 2021 arXiv: 2102.10914.
[26] 王竹溪, 郭敦仁 2012 特殊函数概论 (北京: 北京大学出版社) 第15页
Wang Z X, Guo D R 2012 Special Functions (Beijing: Peking University Press) p15 (in Chinese)
[27] 梁九卿, 韦联福 2011 量子力学新进展 (北京: 科学出版社) 第26页
Liang J Q, Wei L F 2011 New Developments in Quantum Mechanics (Beijing: Science Press) p26 (in Chinese)
[28] Kivshar Y S, Afansjev V V, Snyder A W 1996 Opt. Commun. 126 348
Google Scholar
[29] Triki H, Hamaizi Y, Zhou Q, Biswas A, Ullah M Z, Moshokoa S P, Belic M 2018 Optik 155 329
Google Scholar
[30] Busch T, Anglin J R 2001 Phys. Rev. Lett. 87 010401
Google Scholar
[31] Alejo M A, Corcho A J 2020 arXiv: 2003.09994
[32] Li J D, Meng L Z, Zhao L C 2023 Phys. Rev. A 107 013511
Google Scholar
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图 1 坐标空间中高斯波包线性干涉图样 (a)
$ xz $ 空间两个波包干涉随时间演化; (b)波包中心($ z = 0 $ ) 处波函数密度对时间演化.宽度为a、振幅为S,$ t = 0 $ 时刻位于$ (0, 0) $ 处具有相反x方向动量$\pm k\hat{{\boldsymbol{e}}}_x$ 的高斯波包. 实际参数$S = a = 1, k = 5 $ Figure 1. Linear interference of two Gaussian wave packets in position space: (a) Time evolution of Gaussian wave packet interference in
$ xz $ -space; (b) density plot of wave packet center($ z = 0 $ ) vs. time t. The wave packets of width a and amplitude S start at$ (0, 0) $ with opposite momentum$\pm k\hat{{\boldsymbol{e}}}_x$ in x-direction. The actual parameters are$S = a = 1, k = 5 $ .
图 2 坐标空间中孤子非线性干涉图样 (a)
$ xz $ 空间两个孤子干涉随时间演化； (b)孤子中心($ z = 0 $ ) 处波函数密度对时间演化. 两个具有相反方向速度$ b_1, b_2 $ 的完全相同的孤子. 实际参数为$ a_1 = 1, b_1 = 5, g = 1, a_2 = 1, b_2 = -5, c_1 = d_1 = c_2 = d_2 = 0 $ , 及$ S= 1, a = 1 $ Figure 2. Nonlinear Interference of two solitons in position space: (a) Time evolution of soliton interference in
$ xz $ -space; (b) density plot of soliton center ($ z = 0 $ ) vs. time t. Two identical solitons with opposite velocity$ b_1, b_2 $ . The actual parameters are$ a_1 = 1, b_1 = 5, g = 1, a_2 = 1, b_2 = -5, c_1 = d_1 = c_2 = d_2 = 0 $ , and$ S = 1, a = 1 $ .
图 3
$ t = 0.2 $ 时刻波函数相对相位导数$ \text{d}\phi/\text{d}x $ 重构、磁单极分布及产生的矢势A (a)$ t = 0.2 $ 时刻, 波函数相对相位导数$ \text{d}\phi/\text{d}x $ 解析解与利用磁单极重构解, 黑色实线为解析解, 红色虚线为磁单极重构解; (b)$ t = 0.2 $ 时刻复平面上磁单极分布及对应的矢势A,$ \odot, \otimes $ 分别表示$\mu=\pm\dfrac{1}{2}$ 的两类磁单极,$ \text{Re}[z], \text{Im}[z] $ 分别表示实部虚部. 实际参数同图1Figure 3. Derivative of relative phase function
$ \text{d}\phi/\text{d}x $ , Dirac magnetic monopole distribution and corresponding vector potential A at time$ t = 0.2 $ : (a) Analytic solution and reconstruction using magnetic monopoles of phase function derivative$ \text{d}\phi/\text{d}x $ at time$ t = 0.2 $ , analytic solution(black solid line), construct using magnetic monopole(red dash line); (b) magnetic monopole distribution and corresponding vector potential A on complex plane at time$ t = 0.2 $ ,$ \odot, \otimes $ denotes monopoles with$\mu=\pm\dfrac{1}{2}$ and$ \text{Re}[z], \text{Im}[z] $ real part, imaginary part respectively. The actual parameters are same as Fig. 1.
图 4 复平面上磁单极分布及相应矢势A随时间演化,
$ \odot, \otimes $ 分别表示$\mu=\pm\dfrac{1}{2}$ 的两类磁单极,$ \text{Re}[{\cal{Z}}], \text{Im}[{\cal{Z}}] $ 分别表示实部虚部. 实际参数同图1Figure 4. Time evolution of Magnetic monopole distribution and corresponding vector potential A on complex plane,
$ \odot, \otimes $ denotes monopoles with$\mu=\pm\dfrac{1}{2}$ and$ \text{Re}[{\cal{Z}}], \text{Im}[{\cal{Z}}] $ real part, imaginary part respectively. The actual parameters are same as Fig. 1.
图 5
$ t = 0.05 $ 时刻波函数相对相位导数$ \text{d}\phi/\text{d}x $ 重构、磁单极分布及产生的矢势A (a)$ t = 0.2 $ 时刻, 波函数相对相位导数$ \text{d}\phi/\text{d}x $ 解析解与利用磁单极重构解, 黑色实线为解析解, 红色虚线为磁单极重构解; (b)$ t = 0.2 $ 时刻复平面上磁单极分布及对应的矢势A,$ \odot, \otimes $ 分别表示$\mu=\pm\dfrac{1}{2}$ 的两类磁单极,$ \text{Re}[{\cal{Z}}], \text{Im}[{\cal{Z}}] $ 分别表示实部虚部. 实际参数同图2Figure 5. Derivative of relative phase function
$ \text{d}\phi/\text{d}x $ , Dirac magnetic monopole distribution and corresponding vector potential A at time$ t = 0.05 $ : (a) Analytic solution and reconstruction using magnetic monopoles of phase function derivative$ \text{d}\phi/\text{d}x $ at time$ t = 0.2 $ , analytic solution(black solid line), construct using magnetic monopole(red dash line); (b) magnetic monopole distribution and corresponding vector potential A on complex plane at time$ t = 0.2 $ ,$ \odot, \otimes $ denotes monopoles with$\mu=\pm\dfrac{1}{2}$ and$ \text{Re}[{\cal{Z}}], \text{Im}[{\cal{Z}}] $ real part, imaginary part respectively. The actual parameters are same as Fig. 2.
图 6 复平面上磁单极分布及相应矢势A随时间演化.
$ \odot, \otimes $ 分别表示$\mu=\pm\dfrac{1}{2}$ 的两类磁单极,$ \text{Re}[{\cal{Z}}], \text{Im}[{\cal{Z}}] $ 分别表示实部虚部. 实际参数同图2Figure 6. Time evolution of magnetic monopole distribution and corresponding vector potential A on complex plane,
$ \odot, \otimes $ denotes monopoles with$\mu=\pm\dfrac{1}{2}$ and$ \text{Re}[{\cal{Z}}], \text{Im}[{\cal{Z}}] $ real part, imaginary part respectively. The actual parameters are same as Fig. 2.
图 7 完全碰撞时刻波函数相位
$ \pm\pi $ 跃变与密度零点 (a)高斯波包线性干涉$ t = 0 $ 时刻波函数相位与密度零点; (b)双孤子非线性干涉$ t = 0 $ 时刻波函数相位与密度零点. 线性干涉与非线性干涉情形实际参数分别同图1和图2Figure 7.
$ \pm\pi $ jump of phase function and zeros of density at complete collision time ($ t = 0 $ ): (a) Phase jump and density zeros of Gaussian wave packet linear interference at time$ t = 0 $ ; (b) phase jump and density zeros of double soliton nonlinear interference at time$ t = 0 $ . The actual parameters for linear and nonlinear case are same as Fig. 1 and Fig. 2, respectively. -
[1] Dirac P A M 1931 Proc. R. Soc. Lond. A 133 60
Google Scholar
[2] Milton K A 2006 Rep. Prog. Phys. 69 1637
Google Scholar
[3] Yang C N 1970 Phys. Rev. D 1 2360
Google Scholar
[4] Wu T T, Yang C N 1995 Phys. Rev. D 12 3845
Google Scholar
[5] Berry M V 1980 Eur. J. Phys. 1 240
Google Scholar
[6] Aharonov Y, Bohm D 1959 Phys. Rev. 115 485
Google Scholar
[7] Berry M V 1984 Proc. R. Soc. Lond. A 392 45
Google Scholar
[8] Hooft G 1974 Nucl. Phys. B 79 276
Google Scholar
[9] Castelnovo C, Moessner R, Sondhi S L 2008 Nature 451 42
Google Scholar
[10] Milde P, Köhler D, Seidel J, Eng L M, Bauer A, Chacon A, Kindervater J, Mühlbauer S, Pfleiderer C, Buhrandt S, Schütte C, Rosch A 2013 Science 340 1076
Google Scholar
[11] Ray M W, Ruokokoski E, Kandel S, Möttönen M, Hall D S 2014 Nature 505 657
Google Scholar
[12] Xiao D, Chang M C, Niu Q 2010 Rev. Mod. Phys. 82 1959
Google Scholar
[13] Zhao L C, Qin Y H, Lee C, Liu J 2021 Phys. Rev. E 10 3
Google Scholar
[14] Muga J G, Ruschhaupt A, Campo A 2009 Time in Quantum Mechanics (Vol. 2) (Berlin, Heidelberg: Springer Berlin Heidelberg) p305
[15] Zabusky N J, Kruskal M D 1965 Phys. Rev. Lett. 15 240
Google Scholar
[16] Barenblatt G I 1996 Scaling, Self-similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics (Cambridge: Cambridge University Press)
[17] Karpman V I 1975 Non-Linear Waves in Dispersive Media (New York: Pergamon Press)
[18] Agrawal G 2006 Nonlinear Fiber Optics (Cambridge: Academic Press)
[19] Wu B, Liu J, Niu Q 2002 Phys. Rev. Lett. 88 034101
Google Scholar
[20] Rebbi C, Soliani G 1984 Solitons and Particles (Singapore: World Scientific Publishing)
[21] Nguyen J H V, Dyke P, Luo D, Malomed B A, Hulet R G 2014 Nat. Phys. 10 918
Google Scholar
[22] Zakharov V E, Shabat A B 1973 Sov. Phys. JETP 37 823
[23] Zhao L C, Ling L, Yang Z Y, Liu J 2016 Nonlinear Dyn. 83 659
Google Scholar
[24] Yang C N, Lee T D 1952 Phys. Rev. 87 404
Google Scholar
[25] Zhao L C, Meng L Z, Qin Y H, Yang Z Y, Liu J 2021 arXiv: 2102.10914.
[26] 王竹溪, 郭敦仁 2012 特殊函数概论 (北京: 北京大学出版社) 第15页
Wang Z X, Guo D R 2012 Special Functions (Beijing: Peking University Press) p15 (in Chinese)
[27] 梁九卿, 韦联福 2011 量子力学新进展 (北京: 科学出版社) 第26页
Liang J Q, Wei L F 2011 New Developments in Quantum Mechanics (Beijing: Science Press) p26 (in Chinese)
[28] Kivshar Y S, Afansjev V V, Snyder A W 1996 Opt. Commun. 126 348
Google Scholar
[29] Triki H, Hamaizi Y, Zhou Q, Biswas A, Ullah M Z, Moshokoa S P, Belic M 2018 Optik 155 329
Google Scholar
[30] Busch T, Anglin J R 2001 Phys. Rev. Lett. 87 010401
Google Scholar
[31] Alejo M A, Corcho A J 2020 arXiv: 2003.09994
[32] Li J D, Meng L Z, Zhao L C 2023 Phys. Rev. A 107 013511
Google Scholar
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