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本文构造了耦合自散焦饱和非线性薛定谔方程, 通过改变势函数参数再利用功率守恒的平方算符法, 得到偶极-偶极、三极-偶极以及偶极-三极矢量孤子解. 随着孤子功率的增大, 这3类矢量孤子均能存在, 它们的存在性明显受到势函数的调制. 本文给出了3类矢量孤子由势函数调制的存在区域. 3类矢量孤子的稳定区域受每个分量的孤子功率调制. 随着两分量孤子功率的增大, 3类矢量孤子的稳定域均逐渐扩大. 当饱和非线性强度增大时, 三极-偶极和偶极-三极矢量孤子由稳定状态到不稳定状态临界点对应的孤子功率值逐渐降低. 而偶极-偶极矢量孤子由稳定状态到不稳定状态临界点对应的孤子功率值并不会因为饱和非线性强度增大而变化.
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关键词:
- 耦合非线性薛定谔方程 /
- 多极矢量光孤子 /
- 饱和非线性
We construct the coupled self-defocusing saturated nonlinear Schrödinger equation and obtain the dipole-dipole, tripole-dipole and dipole-tripole vector soliton solutions by changing the potential function parameters and using the square operator method of power conservation. With the increase of soliton power, the dipole-dipole, tripole-dipole and dipole-tripole vector solitons can all exist. The existence of the three kinds of vector solitons is obviously modulated by the potential function. The existence domain of three kinds of vector solitons, modulated by the potential function, is given in this work. The stability domains of three vector solitons are modulated by the soliton power of each component. The stability regions of three kinds of vector solitons expand with the increase of the power of two-component soliton. With the increase of saturation nonlinear strength, the power values of the tripole-dipole and dipole-tripole vector solitons at the critical points from stable state to unstable state decrease gradually, and yet the power of the soliton at the critical point from the stable state to the unstable state does not change.[1] Wang T Y, Zhou Q, Liu W J 2022 Chin. Phys. B 31 020501
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[2] Yan Y Y, Liu W J 2021 Chin. Phys. Lett. 38 094201
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[3] Wang H, Li X, Zhou Q, Liu W 2023 Chaos Soliton. Fract. 166 112924
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[4] Kivshar Y S, Luther-Davies B 1998 Phys. Rep. 298 81
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[5] Cao Q H, Dai C Q 2021 Chin. Phys. Lett. 38 090501
Google Scholar
[6] Bo M X, Tian H P, Li Z H, Zhou G S 2002 J. Quantum 31 030006
[7] Shen Z, Zhang Y L, Chen Y, Sun F W, Zou X B, Guo G C, Zou C L, Dong C H 2018 Nat. Commun. 9 1797
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[8] Su S W, Gou S C, Chew L Y, Chang Y Y, Liao W T 2017 Phys. Rev. A 95 061805
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[9] Xie J, Zhu X, He Y 2019 Nonlinear Dyn. 97 1287
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[10] Christodoulides N D, Joseph I R 1988 Opt. Lett. 13 53
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[11] Desyatnikov A S, Kivshar Y S 2001 Phys. Rev. Lett. 87 033901
Google Scholar
[12] Yang J, De P 2003 Phys. Rev. E 67 016608
Google Scholar
[13] Kartashov Y V, Zelenina A S, Vysloukh V A, Torner L 2004 Phys. Rev. E 70 066623
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[14] Xu Z, Kartashov Y V, Torner L 2006 Phys. Rev. E 73 055601
[15] Kartashov Y V, Torner L, Vysloukh V, Mihalache D 2006 Opt. Lett. 31 1483
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[16] Wang R R, Wang Y Y, Dai C Q 2022 Optik 254 168639
Google Scholar
[17] Bo W B, Liu W, Wang Y Y 2022 Optik 255 168697
Google Scholar
[18] Kartashov Y V 2013 Opt. Lett. 38 2600
Google Scholar
[19] Bo W B, Wang R R, Fang Y, Wang Y Y, Dai C Q 2022 Nonlinear Dyn. 111 1577
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[20] Zhu X, He Y 2018 Opt. Express 26 26511
Google Scholar
[21] Yang J, Lakoba T I 2007 Stud. Appl. Math. 118 153
[22] Li P F, Mihalache D, Malomed, A. B 2018 Philos. Trans. R. Soc. Math. Phys. Eng. Sci. 376 2124
[23] Kartashov Y V, Malomed B A, Torner L 2011 Rev. Mod. Phys. 83 247
Google Scholar
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图 1 (a)
${x_0} = 1.5$ 时势函数图像. 分量${\psi _1}\left( {x, z} \right)$ 孤子功率${P_1}$ 与两分量传播常数${b_{1, 2}}$ 的关系 (b)${x_0} = 1.7$ 时偶极-偶极矢量孤子; (c)${x_0} = 1.6$ 时三极-偶极矢量孤子; (d)${x_0} = 1.5$ 时偶极-三极矢量孤子. 其他参数为${V_0} = 2.8$ ,${w_0} = 1.1$ ,$S = 1$ ,$\sigma = - 1$ ,${P_2} = 3.3$ Fig. 1. (a) Time potential function image when
${x_0} = 1.5$ . The relation between the power of component${\psi _1}\left( {x, z} \right)$ and the propagation constant${b_{1, 2}}$ : (b) dipole - dipole vector soliton when${x_0} = 1.7$ ; (c) tripole-dipole vector solito when${x_0} = 1.6$ ; (d) dipole-tripole vector soliton when${x_0} = 1.5$ . Other parameters are${V_0} = 2.8$ ,${w_0} = 1.1$ ,$S = 1$ ,$\sigma = - 1$ ,${P_2} = 3.3$ .
图 2 (a)
${x_0} = 1.7$ 时偶极-偶极矢量孤子剖面; (b)${x_0} = 1.6$ 时三极-偶极矢量孤子剖面; (c)${x_0} = 1.5$ 时偶极-三极矢量孤子剖面. 其他参数为${V_0} = 2.8$ ,${w_0} = 1.1$ ,$S = 1$ ,$\sigma = - 1$ ,${P_1} = 1.5$ ,${P_2} = 3.3$ Fig. 2. (a) The profile of dipole-dipole vector soliton when
${x_0} = 1.7$ ; (b) the profile of tripole-dipole vector soliton when${x_0} = 1.6$ ; (c) the profile of dipole-tripole vector soliton when${x_0} = 1.5$ . Other parameters are${V_0} = 2.8$ ,${w_0} = 1.1$ ,$S = 1$ ,$\sigma = - 1$ ,${P_1} = 1.5$ and${P_2} = 3.3$ .
图 3 由势函数参数
${V_0}$ ,${x_0}$ 共同调制的(a)偶极-偶极、(b)三极-偶极、(c)偶极-三极孤子存在域. 其他参数为$S = 1$ ,$\sigma = - 1$ ,${w_0} = 1.1$ ,${P_1} = 1.5$ ,${P_2} = 3.3$ Fig. 3. Existence modulated by the potential parameters
${V_0}$ and${x_0}$ of (a) dipole-dipole, (b) tripole-dipole and (c) dipole-tripole solitons. Other parameters are$S = 1$ ,$\sigma = - 1$ ,${w_0} = 1.1$ ,${P_1} = 1.5$ ,${P_2} = 3.3$ .
图 4 (a)偶极-偶极、(b)三极-偶极和(c)偶极-三极孤子的两分量孤子功率
${P_1}$ ,${P_2}$ 共同调制的矢量孤子稳定域. 参数为${V_0} = 2.8$ ,${w_0} = 1.1$ ,$ S = 1 $ ,$\sigma = - 1$ ,${x_0} = 1.7 $ —1.5Fig. 4. Stability modulated by the two-component power
${P_1}$ and${P_2}$ of (a) dipole-dipole, (b) tripole-dipole and (c) dipole-tripole solitons. Parameters are${V_0} = 2.8$ ,${w_0} = 1.1$ ,$S = 1$ ,$\sigma = - 1$ ,${x_0} = 1.7 $ to 1.5.
图 5 不同饱和非线性强度
$S$ 和功率${P_1}$ 值的 (a)偶极-偶极、(b)三极-偶极和 (c)偶极-三极矢量孤子的最大不稳定增长率${\text{Max}}{\delta _{\rm{r}}}$ . 参数为${P_2} = 3$ ,${V_0} = 2.8$ ,${w_0} = 1.1$ ,$\sigma = - 1$ ,${x_0} = 1.7 $ —1.5Fig. 5. The maximum unstable growth rate
${\text{Max}}{\delta _{\rm{r}}}$ of (a) dipole-dipole, (b) tripole-dipole and (c) dipole-tripole vector solitons with different saturated nonlinear strengths$S$ and power values${P_1}$ . Parameters are${P_2} = 3$ ,${V_0} = 2.8$ ,${w_0} = 1.1$ ,$\sigma = - 1$ ,${x_0} = 1.7 $ to 1.5.
图 6 动力学演化图像及相应的特征谱图像 (a), (b)偶极-偶极矢量孤子(
${P_1} = 1, {P_2} = 3$ ); (c), (d)三极-偶极矢量孤子(${P_1} = $ $ 1.5, {P_2} = 3$ ); (e), (f)偶极-三极矢量孤子(${P_1} = 1, {P_2} = 3$ ). 参数为${V_0} = 2.8$ ,${w_0} = 1.1$ ,$S = 1$ ,$\sigma = - 1$ ,${x_0} = 1.7 $ —1.5Fig. 6. Dynamic evolution image and corresponding characteristic spectrum image: (a), (b) Dipole-dipole vector solitons with
${P_1} = 1, {P_2} = 3$ ; (c), (d) tripole-dipole vector soliton with${P_1} = 1.5, {P_2} = 3$ ; (e), (f) dipole-tripole vector solitons with${P_1} = 1, {P_2} = 3$ . Parameters are${V_0} = 2.8$ ,${w_0} = 1.1$ ,$S = 1$ ,$\sigma = - 1$ ,${x_0} = 1.7$ to 1.5. -
[1] Wang T Y, Zhou Q, Liu W J 2022 Chin. Phys. B 31 020501
Google Scholar
[2] Yan Y Y, Liu W J 2021 Chin. Phys. Lett. 38 094201
Google Scholar
[3] Wang H, Li X, Zhou Q, Liu W 2023 Chaos Soliton. Fract. 166 112924
Google Scholar
[4] Kivshar Y S, Luther-Davies B 1998 Phys. Rep. 298 81
Google Scholar
[5] Cao Q H, Dai C Q 2021 Chin. Phys. Lett. 38 090501
Google Scholar
[6] Bo M X, Tian H P, Li Z H, Zhou G S 2002 J. Quantum 31 030006
[7] Shen Z, Zhang Y L, Chen Y, Sun F W, Zou X B, Guo G C, Zou C L, Dong C H 2018 Nat. Commun. 9 1797
Google Scholar
[8] Su S W, Gou S C, Chew L Y, Chang Y Y, Liao W T 2017 Phys. Rev. A 95 061805
Google Scholar
[9] Xie J, Zhu X, He Y 2019 Nonlinear Dyn. 97 1287
Google Scholar
[10] Christodoulides N D, Joseph I R 1988 Opt. Lett. 13 53
Google Scholar
[11] Desyatnikov A S, Kivshar Y S 2001 Phys. Rev. Lett. 87 033901
Google Scholar
[12] Yang J, De P 2003 Phys. Rev. E 67 016608
Google Scholar
[13] Kartashov Y V, Zelenina A S, Vysloukh V A, Torner L 2004 Phys. Rev. E 70 066623
Google Scholar
[14] Xu Z, Kartashov Y V, Torner L 2006 Phys. Rev. E 73 055601
[15] Kartashov Y V, Torner L, Vysloukh V, Mihalache D 2006 Opt. Lett. 31 1483
Google Scholar
[16] Wang R R, Wang Y Y, Dai C Q 2022 Optik 254 168639
Google Scholar
[17] Bo W B, Liu W, Wang Y Y 2022 Optik 255 168697
Google Scholar
[18] Kartashov Y V 2013 Opt. Lett. 38 2600
Google Scholar
[19] Bo W B, Wang R R, Fang Y, Wang Y Y, Dai C Q 2022 Nonlinear Dyn. 111 1577
Google Scholar
[20] Zhu X, He Y 2018 Opt. Express 26 26511
Google Scholar
[21] Yang J, Lakoba T I 2007 Stud. Appl. Math. 118 153
[22] Li P F, Mihalache D, Malomed, A. B 2018 Philos. Trans. R. Soc. Math. Phys. Eng. Sci. 376 2124
[23] Kartashov Y V, Malomed B A, Torner L 2011 Rev. Mod. Phys. 83 247
Google Scholar
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