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Dark soliton molecules in nonlinear optics

Xu Dan-Hong Lou Sen-Yue

Dark soliton molecules in nonlinear optics

Xu Dan-Hong, Lou Sen-Yue
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  • The study on soliton molecules is one of the important topics in nonlinear science especially in nonlinear optics. The bright soliton molecules have been experimentally observed in optics, however, the dark soliton molecules have not yet been experimentally observed. Theoretically, the soliton molecules have been found for some coupled nonlinear systems. Nevertheless, the soliton molecules have not been obtained for non-coupled single component nonlinear models. In this paper, we first study the exact periodic waves (soliton lattices) and solitary waves for a nonlinear nonintegrable optical model with second and third order dispersions and high order nonlinear effects including self-steeping, Raman scattering and nonlinear dispersion. Two types of dark soliton lattice and three types of soliton lattice are explicitly exhibited for general nonintegrable system. Five types of bright (with and without gray background), dark and gray solitons can be obtained from the limit cases of the modules of the soliton lattices. For an integrable case, using a novel generalized bilinear form of a single component nonlinear system, the multi-soliton solutions are obtained and expressed by a completely new form which are invariant under the full reversal transformations such as the parity, the time reversal, the charge conjugate and the field reversal. To find soliton molecules, a novel mechanism, the velocity resonant, is proposed. Starting from the multi-soliton solutions and using the velocity resonance mechanism, the analytical expression of the dark soliton molecules can be readily obtained. For the model given in this paper, the integrable higher order nonlinear Schrodinger equation, one can proved that the interactions among the dark soliton molecules and the usually solitons are elastic. It is worth pointing out that soliton molecules can also exist in the case of nonintegrable systems.
      Corresponding author: Lou Sen-Yue, lousenyue@nbu.edu.cn
    [1]

    Kivshar Y S, Malomed B A 1989 Rev. Mod. Phys. 61 763

    [2]

    Köttig F, Tani T, Travers J C, Russell P St J 2017 Phys. Rev. Lett. 118 263902

    [3]

    Strogatz S 2001 Nature (London) 410 268

    [4]

    Forte S 1992 Rev. Mod. Phys. 64 193

    [5]

    Hertog T, Horowitz G T 2005 Phys. Rev. Lett. 94 221301

    [6]

    Drummond P D, Kheruntsyan K V, He H 1998 Phys. Rev. Lett. 81 3055

    [7]

    Lou S Y, Huang F 2017 Sci. Rep. 7 869

    [8]

    Wright L G, Christodoulides D N, Wise F W 2017 Science 358 94

    [9]

    Dudley J M, Dias F, Erkintalo M, Genty G 2014 Nat. Photonics 8 755

    [10]

    Stratmann M, Pagel T, Mitschke F 2005 Phys. Rev. Lett. 95 143902

    [11]

    Herink G, Kurtz F, Jalali B, Solli D R, Ropers C 2017 Science 356 50

    [12]

    Liu X M, Yao X K, Cui Y D 2018 Phys. Rev. Lett. 121 023905

    [13]

    徐中巍, 张祖兴 2013 物理学报 62 104210

    Xu Z W, Zhang Z X, 2013 Acta Phys. Sin. 62 104210

    [14]

    Sheppard A P, Kivshar Y S 1997 Phys. Rev. E 55 4773

    [15]

    Lakomy K, Nath R, Santos L 2012 Phys. Rev. A 86 013610

    [16]

    Lou S Y 2019 arxiv: 1909.03399 v1[nlin.SI]

    [17]

    Hirota R 1971 Phys. Rev. Lett., 27 1192

    [18]

    Liu S J, Tang X Y, Lou S Y 2018 Chin. Phys. B 27 060201

    [19]

    Ablowitz M J, Clarkson P A 1991 Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge: Cambridge University Press)

    [20]

    Li Y Q, Chen J C, Chen Y, Lou S Y 2014 Chin. Phys. Lett. 31 010201

    [21]

    陈登远 2006 孤子引论 (北京: 科学出版社) 第14−42页

    Chen D Y 2006 Introduction on Solitons (Beijing: China Science Publishing and Media Ltd) pp14−42

    [22]

    Lou S Y 2018 J. Math. Phys. 59 083507

    [23]

    Sasa N, Satsuma J 1991 J. Phys. Soc. Jpn. 60 409

    [24]

    Hirota R 1973 J. Math. Phys. 14 805

    [25]

    Kaup D J, Newell A C 1978 J. Math. Phys. 19 798

    [26]

    楼森岳 2020 物理学报 69 010503

    Lou S Y 2020 Acta Phys. Sin. 69 010503

  • 图 1  由(15) 式描述的亮孤子晶格, 其中参数由(16)式给定

    Figure 1.  Bright soliton lattice described by Eq.(15) with the parameter selected from Eq. (16)

    图 2  由(15)式描述的暗孤子晶格, 其中参数由 (17)式给定

    Figure 2.  Dark soliton lattice described by Eq. (15) with the parameter selected from Eq. (17)

    图 3  由(18)式描述 第二类亮孤子晶格, 其中参数由 (20)式给定

    Figure 3.  Second type of bright soliton lattice described by Eq. (18) with the parameter selected from Eq. (20)

    图 4  第三类亮孤子晶格. 由(21)式描述, 其中参数由 (23)式给定

    Figure 4.  Third type of bright soliton lattice described by Eq. (21) with the parameter selected from Eq. (23)

    图 5  第二类暗孤子晶格由(21)式描述, 其中参数由 (24)式给定

    Figure 5.  Second type of dark soliton lattice described by Eq. (21) with the parameter selected from Eq. (24)

    图 6  由(25)式描述的暗孤子, 其中参数由 (26)式给定

    Figure 6.  Dark soliton described by Eq. (25) with the parameter selected from Eq. (26)

    图 7  由(27)式描述的具有灰背景的亮孤子, 其中参数由 (29)式给定

    Figure 7.  Bright soliton (with gray background) described by Eq. (27) with the parameter selected from Eq. (29)

    图 8  由(27)式描述的暗孤子, 其中参数由 (30)式给定

    Figure 8.  Dark soliton described by Eq. (27) with the parameter selected from Eq. (30)

    图 9  (a)由(27)式描述的暗孤子分子的密度图; (b) 与(a)对应的立体图, 图中参数由 (31)式给定

    Figure 9.  (a)Density plot of the dark soliton molecule described by Eq. (27) with the parameter selected from Eq. (31); (b) three dimensional plot related to Fig.(a)

    图 10  由(42)−(43)式描述的二暗孤子相互作用的密度图, 图中参数由 (44)式给定

    Figure 10.  Density plot of the interaction between two dark solitons described by Eq. (42)and Eq. (43) with the parameter selected from Eq. (44)

    图 11  (a) 由(42)式和(43)式描述的暗孤子分子密度图, 图中参数由 (47)式给定; (b) 与图(a)对应的三维立体图

    Figure 11.  (a) Density plot of the dark soliton molecule described by Eq. (42) and Eq.(43) with the parameter selected from Eq. (47); (b) three dimensional plot related to Fig. (a)

    图 12  由(34)式,(37)式和(38)式描述的暗孤子分子和暗孤子的弹性相互作用的密度图, 图中参数由 (49)式给定

    Figure 12.  Density plot of the interaction between a dark soliton molecule and a dark soliton described by Eq. (34), Eq. (37) and Eq. (38) with the parameter selected from Eq. (49)

    图 13  由(34)式,(37)式和(38)式描述的二暗孤子分子的弹性相互作用的密度图, 图中参数由 (50)式给定

    Figure 13.  Density plot of the interaction between two dark soliton molecules described by Eq. (34), Eq. (37) and Eq. (38) with the parameter selected from Eq. (50)

  • [1]

    Kivshar Y S, Malomed B A 1989 Rev. Mod. Phys. 61 763

    [2]

    Köttig F, Tani T, Travers J C, Russell P St J 2017 Phys. Rev. Lett. 118 263902

    [3]

    Strogatz S 2001 Nature (London) 410 268

    [4]

    Forte S 1992 Rev. Mod. Phys. 64 193

    [5]

    Hertog T, Horowitz G T 2005 Phys. Rev. Lett. 94 221301

    [6]

    Drummond P D, Kheruntsyan K V, He H 1998 Phys. Rev. Lett. 81 3055

    [7]

    Lou S Y, Huang F 2017 Sci. Rep. 7 869

    [8]

    Wright L G, Christodoulides D N, Wise F W 2017 Science 358 94

    [9]

    Dudley J M, Dias F, Erkintalo M, Genty G 2014 Nat. Photonics 8 755

    [10]

    Stratmann M, Pagel T, Mitschke F 2005 Phys. Rev. Lett. 95 143902

    [11]

    Herink G, Kurtz F, Jalali B, Solli D R, Ropers C 2017 Science 356 50

    [12]

    Liu X M, Yao X K, Cui Y D 2018 Phys. Rev. Lett. 121 023905

    [13]

    徐中巍, 张祖兴 2013 物理学报 62 104210

    Xu Z W, Zhang Z X, 2013 Acta Phys. Sin. 62 104210

    [14]

    Sheppard A P, Kivshar Y S 1997 Phys. Rev. E 55 4773

    [15]

    Lakomy K, Nath R, Santos L 2012 Phys. Rev. A 86 013610

    [16]

    Lou S Y 2019 arxiv: 1909.03399 v1[nlin.SI]

    [17]

    Hirota R 1971 Phys. Rev. Lett., 27 1192

    [18]

    Liu S J, Tang X Y, Lou S Y 2018 Chin. Phys. B 27 060201

    [19]

    Ablowitz M J, Clarkson P A 1991 Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge: Cambridge University Press)

    [20]

    Li Y Q, Chen J C, Chen Y, Lou S Y 2014 Chin. Phys. Lett. 31 010201

    [21]

    陈登远 2006 孤子引论 (北京: 科学出版社) 第14−42页

    Chen D Y 2006 Introduction on Solitons (Beijing: China Science Publishing and Media Ltd) pp14−42

    [22]

    Lou S Y 2018 J. Math. Phys. 59 083507

    [23]

    Sasa N, Satsuma J 1991 J. Phys. Soc. Jpn. 60 409

    [24]

    Hirota R 1973 J. Math. Phys. 14 805

    [25]

    Kaup D J, Newell A C 1978 J. Math. Phys. 19 798

    [26]

    楼森岳 2020 物理学报 69 010503

    Lou S Y 2020 Acta Phys. Sin. 69 010503

  • [1] Lou Sen-Yue. Full reversal symmetric multiple soliton solutions for integrable systems. Acta Physica Sinica, 2020, 69(1): 010503. doi: 10.7498/aps.69.20191172
    [2] Li Bo, Wang Yan-Shen. The q-boson hopping model with integrable open boundary condition. Acta Physica Sinica, 2007, 56(3): 1260-1265. doi: 10.7498/aps.56.1260
    [3] Yan Qing-You, Zhang Yu-Feng. A type of expanding integrable system for NLS-mKdV hierarchy. Acta Physica Sinica, 2003, 52(9): 2109-2113. doi: 10.7498/aps.52.2109
    [4] Wei Qing, Wang Qi, Shi Jie-Long, Chen Yuan-Yuan. . Acta Physica Sinica, 2002, 51(1): 99-103. doi: 10.7498/aps.51.99
    [5] RUAN HANG-YU. STUDY OF SOLITONS INTERACTION IN INTEGRABLE MODELS. Acta Physica Sinica, 2001, 50(3): 369-376. doi: 10.7498/aps.50.369
    [6] Li Qi-Liang, Zhu Hai-Dong, Tang Xiang-Hong, Li Cheng-Jia, Wang Xiao-Jun, Lin Li-Bin. Integrability aspects of solitons’ coupled equation in multi-wavelength system. Acta Physica Sinica, 2004, 53(6): 1623-1628. doi: 10.7498/aps.53.1623
    [7] Jiang De-Sheng, Ouyang Shi-Gen, She Wei-Long. Interaction of photovotaic spatial solitons between dark and dark as well as dark and bright. Acta Physica Sinica, 2004, 53(11): 3777-3785. doi: 10.7498/aps.53.3777
    [8] He Xian-tu. NON-LINEAR EFFECT ON THE LARGE AMPLITUDE WAVES INTERACTION WITH PARTICLES OF LOW FREQUENCY OSCILLATION IN PLASMA. Acta Physica Sinica, 1982, 31(10): 1317-1336. doi: 10.7498/aps.31.1317
    [9] Yu Wen-Jian, Wang Ji-Suo, Liang Bao-Long. Quantum properties of two-level atoms interacting with nonlinear coherent states. Acta Physica Sinica, 2012, 61(6): 060301. doi: 10.7498/aps.61.060301
    [10] Xiao Qi, Wang Jun, Guo Xia-Sheng, Zhang Dong. Nonlinear interaction between ultrasound and solid rough interface. Acta Physica Sinica, 2013, 62(9): 094301. doi: 10.7498/aps.62.094301
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  • Received Date:  06 September 2019
  • Accepted Date:  25 October 2019
  • Available Online:  12 December 2019
  • Published Online:  01 January 2020

Dark soliton molecules in nonlinear optics

    Corresponding author: Lou Sen-Yue, lousenyue@nbu.edu.cn
  • School of Physical Science and Technology, Ningbo University, Ningbo 315211, China

Abstract: The study on soliton molecules is one of the important topics in nonlinear science especially in nonlinear optics. The bright soliton molecules have been experimentally observed in optics, however, the dark soliton molecules have not yet been experimentally observed. Theoretically, the soliton molecules have been found for some coupled nonlinear systems. Nevertheless, the soliton molecules have not been obtained for non-coupled single component nonlinear models. In this paper, we first study the exact periodic waves (soliton lattices) and solitary waves for a nonlinear nonintegrable optical model with second and third order dispersions and high order nonlinear effects including self-steeping, Raman scattering and nonlinear dispersion. Two types of dark soliton lattice and three types of soliton lattice are explicitly exhibited for general nonintegrable system. Five types of bright (with and without gray background), dark and gray solitons can be obtained from the limit cases of the modules of the soliton lattices. For an integrable case, using a novel generalized bilinear form of a single component nonlinear system, the multi-soliton solutions are obtained and expressed by a completely new form which are invariant under the full reversal transformations such as the parity, the time reversal, the charge conjugate and the field reversal. To find soliton molecules, a novel mechanism, the velocity resonant, is proposed. Starting from the multi-soliton solutions and using the velocity resonance mechanism, the analytical expression of the dark soliton molecules can be readily obtained. For the model given in this paper, the integrable higher order nonlinear Schrodinger equation, one can proved that the interactions among the dark soliton molecules and the usually solitons are elastic. It is worth pointing out that soliton molecules can also exist in the case of nonintegrable systems.

1.   引 言
  • 孤立波、孤立子、呼吸子、怪波等非线性局域激发模式在物理学的各个分支如流体物理[1]、等离子体物理[2]、复杂系统和复杂网络[3]、量子场论和粒子物理[4]、引力理论[5]、玻色爱恩斯坦凝聚[6]、大气和海洋物理[7]、特别是光物理[8-10] 中起着非常重要的作用. 最近, 孤子分子在非线性光学实验上的成功发现[11-13] 成为非线性物理的一个新的热门课题. 在理论上, 非线性耦合系统的孤子分子已经被一些非线性物理工作者所研究[14,15]. 除了在光学中的孤子分子外, 在其他领域也应该能找到孤子分子. 在文献[16]中, 本文作者之一(楼)在单分量的流体模型中给出了多种类型的孤子分子解.

    非线性系统孤子解的求解有很多行之有效的方法, 如广田(Hirota)直接法[17]、达布(Darboux)变换方法[18]、反散射变换方法[19]、对称性方法[20]等等. 通常使用不同的方法得到的多孤子解表面上看可以是很不一样的, 但本质上都是等价的. 不同的表达式在应用中各有优点. 绝大多数文献中, 各种非线性模型的单孤子解都采用紧致简洁方便的双曲函数形式, 因此很多著名专家如Hirota 和Toda 及我国的陈登远[21]等都期望能用双曲函数来简洁地表达多孤子解, 最近, 我们成功实现这一愿望[7,22]. 这些新的简洁表达式不仅研究非局域非线性系统的孤子解非常方便, 而且本文将进一步用这种表达式来寻求各种类型的共振孤子, 包括孤子分子.

    本文第二节首先给出具有高阶色散和高阶非线性修正的非线性光学系统. 然后研究该系统可能具有的周期波解和孤立波解. 本文第三节中研究一个可积的情况—散焦型Hirota 模型, 将Hirota模型变换成一个新的Hirota 双线性方程后, 成功给出用双曲余弦描述的多暗孤子表达式. 然后从多孤子解出发研究一种新的共振孤子激发模式—速度共振. 从而导致孤子束缚态—暗孤子分子的形成. 暗孤子分子之间及暗孤子分子和通常暗孤子的相互作用性质也在第三节做了简单的讨论. 最后一节是总结和讨论.

    • 在众多的实际非线性光学问题中高阶的色散和各种非线性效应是不可忽略的. 其中最常见的高阶非线性薛定谔方程可具有下述形式($ \rm{i} = \sqrt{-1} $)

      方程(1)包含了很多物理和数学领域中重要的特殊情况(包括时空变量$ t $$ x $的交换). $ \epsilon = 0 $ 对应的正是著名的聚焦($ \sigma = 1 $) 和散焦($ \sigma = -1 $)非线性薛定谔方程. 当$ d_2 = r_2 = 0, \ \beta = 0, \ 3 $ 时, 方程(1) 是另一个非常重要的物理和数学中的重要模型: 复修正Korteweg de-Vrise (KdV) 方程[23]. 当$ \beta = 0 $时, 方程(1)是可积的Hirota 方程[24]. 当$ \beta = 3 $时, 方程(1)也是可积系统: Sasa-Satsuma方程[23]. 当$ r_2 = d_3 = 0 $时, 方程(1)也包含了三种可积的导数非线性薛定谔方程[25].

      在研究高阶非线性薛定谔方程的多孤子解和孤子分子解之前, 我们先研究一般高阶非线性薛定谔系统(1)式 下述形式的一般行波解

      其中 $ k_0, \ \omega_0, \ \phi_0, \ k, \ \omega $$ \xi_0 $ 为待定实常数.

      将(2) 式代入(1)式可得关于$ Q(\xi) $应满足的两个常微分方程,

      为了给出(3)式和(4)式的相容显式解我们分两种情况:

      情况1 一般情况. 在对模型参数无任何限制的情况下, 可取载波参数$ k_0 $$ \omega_0 $由其他参数决定为

      在参数条件(5)式和(6)式下, 方程(3)和(4)退化成单个标准的Jacobi椭圆函数满足的方程

      方程(7)有三种标准的周期波解:

      周期波解1

      周期波解2

      周期波解3

      当Jacobi椭圆函数的模趋向于1时, 上述三类周期波解退化成两种类型的孤立子解.

      孤子解1, 暗孤子解

      孤子解2, 亮孤子解

      从解的表达式很容易看出, 暗孤子晶格解(8)式和暗孤子解(11) 式的存在条件是$ \dfrac{\sigma d_3}{r_3(3+\beta)} < 0 $. 亮孤子晶格(9)式, (10)式 和亮孤子(12)式存在条件是$ \dfrac{\sigma d_3}{r_3(3+\beta)} > 0 $.

      情况2 特殊情况$ \varDelta = 0 $. 在这一特殊参数条件下光孤子系统(1)具有更一般的孤子晶格和孤子解. 当$ \varDelta = 0 $ 时, 可简单取

      在参数条件(13)式下, 方程(3)和(4)退化成单个三阶方程

      类似地, 方程(14)有三种标准的周期波(孤子晶格)解:

      周期波解4

      对于小的$ m $, Jacobi椭圆函数sn接近于三角函数sin. 而随着模m 越来越接近于1, sn越来越接近于双曲tanh函数, 而相应的周期解就看起来像是孤立子的周期性排列. 因此对于接近于m = 1的周期解, 也可以称之为孤子晶格解. 图1 展示了用(15) 式描述的周期波解、亮孤子晶格解. 其相应的参数为(本节所有的图中的模型参数固定为$ \beta = d_2 = d_3 = r_2 = 10\epsilon = 1 $)

      Figure 1.  Bright soliton lattice described by Eq.(15) with the parameter selected from Eq. (16)

      图2展示了用同一表达式(15)式 描述的暗孤子晶格解. 其相应的参数为

      Figure 2.  Dark soliton lattice described by Eq. (15) with the parameter selected from Eq. (17)

      这类暗孤子晶格中的暗孤子中间有一非暗的灰色区.

      周期波解5

      图3 展示了第二种类型的亮孤子晶格结构. 这类亮孤子晶格由表达式(18) 式描述. 与图3对应的参数为

      Figure 3.  Second type of bright soliton lattice described by Eq. (18) with the parameter selected from Eq. (20)

      第一类的亮孤子晶格的背景是暗的, 第二类亮孤子晶格的背景是灰的, 而灰色背景和亮孤子间又有暗区相间.

      周期波解6

      由于Jacibi椭圆函数dn的恒正性, 这类函数描述的孤子晶格既可以是亮孤子晶格, 也可以是暗孤子晶格.

      图4 展示了第三种类型的亮孤子晶格结构. 这类亮孤子晶格由表达式(21)式 描述. 与图4对应的参数为

      Figure 4.  Third type of bright soliton lattice described by Eq. (21) with the parameter selected from Eq. (23)

      这一类亮孤子也具有灰色背景, 但不同于第二类亮孤子晶格, 灰背景和亮孤子之间没有暗区隔离.

      图5展示了第二类暗孤子晶格结构. 这类暗孤子晶格由表达式(21) 式描述. 与图5对应的参数为

      Figure 5.  Second type of dark soliton lattice described by Eq. (21) with the parameter selected from Eq. (24)

      当Jacobi椭圆函数的模$ m\rightarrow 1 $时, 周期波解(15)退化为下述孤子解:

      孤子解3

      暗孤子解 (25) 式的实条件和非奇异条件为

      图6正是与(25)式对应的暗孤子解, 相应的参数为

      Figure 6.  Dark soliton described by Eq. (25) with the parameter selected from Eq. (26)

      周期波解(18)式和(21)式退化成同一种孤子解:

      孤子解4:

      当参数满足下述条件

      时, 孤子解 (27) 式是一个具有灰背景的亮孤子.

      图7展示的正是由(27)式表示的具有灰背景的亮孤子, 其中的参数为

      Figure 7.  Bright soliton (with gray background) described by Eq. (27) with the parameter selected from Eq. (29)

      当下述条件

      满足时, 孤子解 (27) 式是一个灰孤子解. 当 $ \beta_1 = $$2\alpha_1 $时, 孤子解 (27) 式是一个暗孤子.

      图8展示了由(27)式表示的暗孤子, 其中的参数取为

      Figure 8.  Dark soliton described by Eq. (27) with the parameter selected from Eq. (30)

      图8所示的暗孤子具有一特殊的性质, 即在暗孤子中心$ x = 0 $ 前三阶变化率均为零: $ I(x = 0) = $$I_{x}(x = 0) =I_{xx}(x = 0) = I_{xxx}(x = 0) = 0 $.

      特别有意义的是表达式(27)式具有共振孤子相同的形式, 因此对于某些参数区间可知(27) 式也具有暗孤子分子的形式. 图9显示了由(27)式描述的孤子分子的性质, 其中的参数选择为

      Figure 9.  (a)Density plot of the dark soliton molecule described by Eq. (27) with the parameter selected from Eq. (31); (b) three dimensional plot related to Fig.(a)

      由于$ \beta\neq 0, \ 3 $时, (1)式不是可积系统, 所以孤子分子可以在不可积系统中存在.

    3.   散焦Hirota系统的多孤子解及其孤子分子解
    • 为了研究孤子分子的性质, 本节只限定于一个特殊的可积系统, Hirota系统, 其对应的参数限制为

      并引入新标记$ \epsilon_1\equiv \epsilon d_3 d_2^{-1} $. 在参数条件(32)式下, 模型方程(1)变为

      对于可积系统, 可以用很多方法得到其多孤子解, 如Hirota 方法、反散射方法、黎曼-希尔伯特方法、达布变换方法等等. 所有各种方法中, Hirota方法是最简单明了的方法.

      很容易验证, Hirota系统在变换

      下可以变为其双线性形式

      在传统的非线性薛定谔模型($ \epsilon_1 = 0 $)和Hirota系统($ \epsilon_1\neq 0 $)的双线性形式中, (35)式 和(36) 式中常数$ \kappa $被固定为零. 当$ \kappa $ 取零时双线性方程组(35) 式和 (36)式的多孤子解由传统的Hirota形式描述. 当$ \kappa\neq0 $${\sigma d_2}/{r_2} < 0 $(对应于散焦系统)时, 双线性系统 (35)式和(36)式的多孤子解可以表示为[26]

      其中关于$ \{\nu\}\equiv \{\nu_1, \ \cdots, \ \nu_N\} $ 的求和是关于所有可能的非对偶分布$ \nu_i = \pm 1, \ i = 1, \ \cdots, \ N $求和. 如果$ \{\nu'\} = -\{\nu\} $, 则分布$ \{\nu'\} $$ \{\nu\} $被称作是对偶的. 由于$ \cosh $函数是偶函数, 所以对偶的分布给出相同的表达式, 所以求和只需要对非对偶分布求. 在(37)式和(38)式中的$ K_{\{\nu\}} $

      $ k_i $, $ \omega_0 $$ \omega_i $

      $ k_0, \ \xi_{i0}, \ \phi_i $$ \phi_0 $ 为任意常数.

      为了寻求孤子分子解, 我们先显式写下双孤子解,

      在一般情况下双孤子解(42)式和(43)式是具有弹性相互作用的双孤子态. 图10展示了这样一个典型的双孤子作用图像, 其中采用的参数为(本节中统一采用的模型参数是$ d_2 = r_2 = -\sigma = 10\epsilon_1 = 1 $)

      Figure 10.  Density plot of the interaction between two dark solitons described by Eq. (42)and Eq. (43) with the parameter selected from Eq. (44)

      只有当二孤子的波速达到一种特殊共振状态, 使得孤子的吸引和排斥作用正好达到平衡时, 才能形成束缚态——孤子分子.

      从表达式(41)式可知两孤子形成孤子分子的速度共振条件为

      图11展示了孤子分子对应的光强的密度图(图11(a))和立体图(图11(b)), 与图对应的参数为

      Figure 11.  (a) Density plot of the dark soliton molecule described by Eq. (42) and Eq.(43) with the parameter selected from Eq. (47); (b) three dimensional plot related to Fig. (a)

      由于可积系统的多孤子之间的相互作用是弹性的, 孤子分子的存在只对应于一些特殊的多孤子, 自然孤子分子和通常的孤子的相互作用及孤子分子之间的相互作用也是弹性的.

      (33)式的多暗孤子分子和多暗孤子的混合解由 (34)式, (37)式和(38)式统一表达, 其中若有$ n $对孤子的速度达到共振条件

      则解(34)式, (37)式和(38)式表示了$ n $暗孤子分子和$ N-2 n $ 暗孤子的混合相互作用解.

      图12展示的是一个暗孤子分子和一个暗孤子的相互作用. 图中对应的参数为

      Figure 12.  Density plot of the interaction between a dark soliton molecule and a dark soliton described by Eq. (34), Eq. (37) and Eq. (38) with the parameter selected from Eq. (49)

      图13展示了二暗孤子分子的相互作用. 图中对应的参数为

      Figure 13.  Density plot of the interaction between two dark soliton molecules described by Eq. (34), Eq. (37) and Eq. (38) with the parameter selected from Eq. (50)

    4.   结论和讨论
    • 本文首先系统研究了一个非线性光学中包含三阶色散、自陡峭效应和自激Raman散射等高阶非线性色散效应的一般非线性薛定谔方程的包络行波解. 结果发现高阶非线性薛定谔系统具有非常丰富的周期波模式, 包括了多种亮孤子晶格(暗背景和灰背景的亮孤子晶格)和暗孤子晶格(双谷暗孤子晶格和单谷暗孤子晶格). 孤子晶格的多样性导致了孤立子(孤子晶格周期解趋于无穷时的解)的多样性: (暗背景)亮孤子, 具有灰背景的亮孤子, 暗孤子, 灰孤子, 超平暗孤子(谷底一、二、三阶变化率为零, 因此也可称之为扭结-反扭结分子), 双暗孤子分子等等.

      对于一个可积的非线性光学系统——散焦型Hirota模型, 得到了一个新的双线性形式, 利用这一新的双线性形式, 给出了一个多孤子解的用双曲余弦描述的紧致形式. 从这一多孤子解出发, 引入速度共振条件即可得到多暗孤子分子和多暗孤子的混合解. 暗孤子分子之间的相互作用以及暗孤子分子和暗孤子之间的相互作用是弹性相互作用.

      孤子分子是当前非线性科学中的重要课题之一, 它象通常的孤子一样也可以在物理学的各个领域中得到应用. 在非线性系统中还存在各种各样的局域激发模式, 如呼吸子、拱形(dromion) 解、团块(lump) 解、尖峰子(peakon)解和紧子(compacton)解等等. 因此自然可以期待得到各种各样的其他类型的分子解, 如呼吸子分子、呼吸子-孤子分子、dromion分子、lump分子、dromion-lump 分子、尖峰子分子、紧子分子等等. 所有有关这些新类型的非线性局域激发的分子解及其可能的物理应用将在以后进行深入的研究.

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