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铁磁纳米线中磁化强度的磁怪波

李再东 郭奇奇

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铁磁纳米线中磁化强度的磁怪波

李再东, 郭奇奇

Rogue wave solution in ferromagnetic nanowires

Li Zai-Dong, Guo Qi-Qi
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  • 本文介绍了铁磁纳米线中磁化强度的一些新激发态, 包括各向同性铁磁的Akhmediev呼吸子、Kuznetsov-Ma孤子和怪波、自旋极化电流驱动下各向异性铁磁纳米线中的怪波动力学. 在各向同性情况下, 展示了形如四片花瓣的磁孤子的空间周期过程和自旋波背景的局域化过程; 在极限情况下, 得到了磁怪波解并阐明了其形成机制. 在各向异性情况下, 发现怪波的产生主要源于中心的能量积累和快速弥散; 此外, 怪波还具有不稳定性, 它和自旋波背景间的能量与磁振子的交换可以通过自旋极化电流来调控.
    In this paper, we introduce some new excited states of magnetization in ferromagnetic nanowires, including Akhmediev breathers, Kuznetsov-Ma soliton and rogue wave in isotropic ferromagnetic nanowires, and rogue wave in anisotropic ferromagnetic nanowires driven by spin-polarized current. The isotropic case demonstrates a spatial periodic process of a magnetic soliton forming the petal with four pieces and a localized process of the spin-wave background. In a limit case, we get rogue waves and clarify its formation mechanism. In the case of anisotropy, it is found that the generation of rogue waves mainly comes from the accumulation of energy and rapid dispersion in the center. In addition, rogue waves are unstable, the spin-polarized current can control the exchange rate of magnons between the envelope soliton and the background. These results can be useful for the exploration of nonlinear excitation in Bosonic and fermionic ferromagnet.
      通信作者: 李再东, lizd@hebut.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 61774001)和河北省自然科学基金(批准号: F2019202141)资助的课题
      Corresponding author: Li Zai-Dong, lizd@hebut.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61774001) and the Natural Science Foundation of Hebei Province, China (Grant No. F2019202141)
    [1]

    Bu K M, Kwon H Y, Kang S P, Kim H J, Won C 2013 J. Magn. Magn. Mater. 343 32Google Scholar

    [2]

    Moon J H, Seo S M, Lee K J, Kim K W, Ryu J, Lee H W, McMichael R D, Stiles M D 2013 Phys. Rev. B 88 184404Google Scholar

    [3]

    Yu X Z, Kanazawa N, Onose Y, Kimoto K, Zhang W Z, Ishiwata S, Matsui Y, Tokura Y 2011 Nat. Mater. 10 106Google Scholar

    [4]

    Rohart S, Thiaville A 2013 Phys. Rev. B 88 184422Google Scholar

    [5]

    Robler U K, Bogdanov A N, Pfleiderer C 2006 Nature 442 797Google Scholar

    [6]

    Allwood D A, Xiong G, Faulkner C C, Atkinson D, Petit D, Cowburn R P 2005 Science 309 1688Google Scholar

    [7]

    Parkin S S P, Hayashi M, Thomas L 2008 Science 320 190Google Scholar

    [8]

    Li Z D, Hu Y C, He P B, Sun L L 2018 Chin. Phys. B 27 077505Google Scholar

    [9]

    He P B, Xie X C, Liu W M 2005 Phys. Rev. B 72 172411Google Scholar

    [10]

    Tretiakov O A, Liu Y, Abanov Ar 2012 Phys. Rev. Lett. 108 247201Google Scholar

    [11]

    Li Q Y, Zhao F, He P B, Li Z D 2015 Chin. Phys. B 24 037508Google Scholar

    [12]

    Kosevich A M, Ivanov B A, Kovalev A S 1990 Phys. Rep. 194 117Google Scholar

    [13]

    Mikeska H J, Steiner M 1991 Adv. Phys. 40 191Google Scholar

    [14]

    Haazen P P J, Mure E, Franken J H, Lavrijsen R, H. Swagten J M, Koopmans B 2013 Nat. Mater. 12 299Google Scholar

    [15]

    Miron I M, Gaudin G, Auffret S, Rodmacq B, Schuhl A, Pizzini S, Vogel J, Gambardella P 2010 Nat. Mater. 9 230Google Scholar

    [16]

    Di K, Zhang V L, Lim H S, Ng S C, Kuok M H, Yu J, Yoon J, Qiu X, Yang H 2015 Phys. Rev. Lett. 114 047201Google Scholar

    [17]

    Wang W W, Albert M, Beg M, Bisotti M A, Chernyshenko D 2015 Phys. Rev. Lett. 114 087203Google Scholar

    [18]

    Li Z D, Cui H, Li Q Y, He P B 2018 Ann. Phys. 388 390Google Scholar

    [19]

    Chesi S, Coish W A 2015 Phys. Rev. B 91 245306Google Scholar

    [20]

    Li Z D, Liu F, Li Q Y, He P B 2015 J. Appl. Phys. 117 173906Google Scholar

    [21]

    Pfleiderer C, Julian S R, Lonzarich G G 2001 Nature 414 427Google Scholar

    [22]

    Uchida M, Onose Y, Matsui Y, Tokura Y 2006 Science 311 359Google Scholar

    [23]

    Meckler S, Mikuszeit N, Preßler A, Vedmedenko E Y, Pietzsch O, Wiesendanger R 2009 Phys. Rev. Lett. 103 157201Google Scholar

    [24]

    Moriya T 1960 Phys.Rev. 120 91Google Scholar

    [25]

    Zakharov D V, Deisenhofer J 2006 Phys. Rev. B 73 094452Google Scholar

    [26]

    Gangadharaiah S, Sun J M, Starykh O A 2008 Phys. Rev. B 78 054436Google Scholar

    [27]

    Albert F J, Emley N C, Myers E B, Ralph D C, Buhrman R A 2002 Phys. Rev. Lett. 89 226802Google Scholar

    [28]

    Mucciolo E R, Chamon C, Marcus C M 2002 Phys. Rev. Lett. 89 146802Google Scholar

    [29]

    Beach G S D, Knutson C, Nistor C, Tsoi M, Erskine J L 2006 Phys. Rev. Lett. 97 057203Google Scholar

    [30]

    Bertotti G, Serpico C, Mayergoyz I D, Magni A, Aquino M, Bonin R 2005 Phys. Rev. Lett. 94 127206Google Scholar

    [31]

    Garcia-Sanchez F, Borys P, Soucaille R, Adam J P, Stamps R L, Kim J V 2015 Phys. Rev. Lett. 114 247206Google Scholar

    [32]

    Katine J A, Albert F J, Buhrman R A, Myers E B, Ralph D C 2000 Phys. Rev. Lett. 84 3149Google Scholar

    [33]

    Tsoi M, Jansen A G M, Bass J, Chiang W C, Seck M, Tsoi V, Wyder P 1998 Phys. Rev. Lett. 80 4281Google Scholar

    [34]

    He P B, Liu W M 2005 Phys. Rev. B 72 064410Google Scholar

    [35]

    Li Z D, He P B, Liu W M 2014 Chin. Phys. B 23 117502Google Scholar

    [36]

    Tsoi M, Tsoi V, Bass J, Jansen A G M, Wyder P 2002 Phys. Rev. Lett. 89 246803Google Scholar

    [37]

    Tserkovnyak Y, Brataas A, Bauer G E W 2002 Phys. Rev. Lett. 88 117601Google Scholar

    [38]

    Li Z D, Li Q Y, He P B, Liang J Q, Liu W M, Fu G S 2010 Phys. Rev. A 81 015602Google Scholar

    [39]

    Freimuth F, Blugel S, Mokrousov Y 2014 Phys. Rev. B 90 174423Google Scholar

    [40]

    Santos T S, Lee J S, Migdal P, Lekshmi I C, Satpati B, Moodera J S 2007 Phys. Rev. Lett. 98 016601Google Scholar

    [41]

    Li Z, Zhang S 2004 Phys. Rev. Lett. 92 207203Google Scholar

    [42]

    Kasai S, Nakatani Y, Kobayashi K, Kohno H, Ono T 2006 Phys. Rev. Lett. 97 107204Google Scholar

    [43]

    Kasai S, Fischer P, Im M Y, Yamada K, Nakatani Y, Kobayashi K, Kohno H, Ono T 2008 Phys. Rev. Lett. 101 237203; Sugimoto S, Fukuma Y, Otani Y C 2011 IEEI. T. Magn. 47 2951

    [44]

    Yamada K, Kasai S, Nakatani Y, Kobayashi K, Kohno H, Thiaville A, Ono T 2007 Nat. Mater. 6 270; Moriya R, Thomas L, Hayashi M, Bazaliy Y B, Rettner C, Parkin S S P 2008 Nat. Phys. 4 368

    [45]

    Zhao L C, Ling L 2016 J. Opt. Soc. Am. B 33 850Google Scholar

    [46]

    Liu C, Yang Z Y, Zhao L C, Yang W L 2015 Phys. Rev. E 91 022904Google Scholar

    [47]

    Duan L, Zhao L C, Xu W H 2017 Phys. Rev. E 95 042212Google Scholar

    [48]

    Wang L, X Wuan, Zhang H Y 2018 Phys. Lett. A 382 2650Google Scholar

    [49]

    Wang L, Liu C, Wu X, Wang X, Sun W R 2018 Nonlinear Dyn. 94 977Google Scholar

    [50]

    Zakharov V E, Gelash A A 2013 Phys. Rev. Lett. 111 054101Google Scholar

    [51]

    Gelash A A, Zakharov V E 2014 Nonlinearity. 27 R1Google Scholar

    [52]

    Yan P, Wang X S, Wang X R 2011 Phys. Rev. Lett. 107 177207Google Scholar

    [53]

    Zhao F, Li Z D, Li Q Y, Wen L, Fu G S, Liu W M 2012 Ann. Phys. 327 2085Google Scholar

    [54]

    Li Z D, Li Q Y, Xu T F, He P B 2016 Phys. Rev. E 94 042220Google Scholar

    [55]

    Li Q Y, Li Z D, He P B, Song W W, Fu G S 2010 Can. J. Phys. 88 9

    [56]

    Hasegawa A 1984 Opt. Lett. 9 288; Tai K, Tomita A, Jewell J L, Hasegawa A 1986 Appl. Phys. Lett. 49 236

    [57]

    Akhmediev N N 2001 Nature. 413 267; Van Simaeys G, Emplit P, Haelterman M 2001 Phys. Rev. Lett. 87 033902; Mussot A, Kudlinski A, Droques M, Szriftgiser P, Akhmediev N 2014 Phys. Rev. X 4 011054

    [58]

    Zhang H Q, Tian B, Xing L, Meng X H 2010 Physica A 389 367Google Scholar

    [59]

    Matveev V B, Salli M A 1991 Darboux Transformations and Solitons, Vol. 5 (Berlin: Springer) pp7−15

    [60]

    谷超豪, 胡和生, 周子翔 2005 孤立子理论中的达布变换及其几何应用(第二版)(上海: 上海科学技术出版社)第18−24页

    Gu C H, Hu H S, Zhou Z X 2005 Darboux Transformation in Soliton Theory and Its Geometric Applications (Shanghai: Scientific and Technical Publishers) pp18−24 (in Chinese).

    [61]

    Li Z D, Wu X, Li Q Y, He P B 2016 Chin. Phys. B 25 010507Google Scholar

    [62]

    Li Z D, Huo C Z, Li Q Y, He P B, Xu T F 2018 Chin. Phys. B 27 040505Google Scholar

    [63]

    Ho T L 1998 Phys. Rev. Lett. 81 742; Law C K, Ohmi T, Machida K, 1998 J. Phys. Soc. Jpn. 67 1822; Law C K, Pu H, Bigelow N P 1998 Phys. Rev. Lett. 81 5257

    [64]

    Pu H, Zhang W P, Meystre P 2001 Phys. Rev. Lett. 87 140405Google Scholar

    [65]

    Li Z D, He P B, Li L, Liang J Q, Liu W M 2005 Phys. Rev. A 71 053611Google Scholar

  • 图 1  方程(3)在$\mu \to {A_{\rm{s}}}{k_{\rm{s}}}, \nu = {k_{\rm{s}}}\sqrt {1 - A_{\rm{s}}^2} $的极限条件下, 磁化分量${m_3}$的渐近过程, 参数如下: As = 0.9, ks = 1 (a) μ = 0.82; (b) μ = 0.89; (c) μ = 0.89999; (d) μ = 1.1; (e) μ = 0.96; (f) μ = 0.9001

    Fig. 1.  The asymptotic processes of the magnetic component ${m_3}$ in the limit processes $\mu \to {A_{\rm{s}}}{k_{\rm{s}}}$ and $\nu = {k_{\rm{s}}}\sqrt {1 - A_{\rm{s}}^2} $ in Eq. (3), where the parameters are as follows: As = 0.9, ks = 1: (a) μ = 0.82; (b) μ = 0.89; (c) μ = 0.89999; (d) μ = 1.1; (e) μ = 0.96; (f) μ = 0.9001, respectively.

    图 2  方程(3)和(7)中磁化强度${{m}} = \left( {{m_1}, {m_2}, {m_3}} \right)$的怪波演化图, 即(a)−(c)为亮怪波, (d)−(f)为暗怪波. 参数: ${A_{\rm{s}}} = \sqrt 3 /2, {k_{\rm{s}}} = 1.5, \nu = {k_{\rm{s}}}\sqrt {1 - A_{\rm{s}}^2}, \mu = \pm 0.75 \sqrt 3 $, ±分别表示亮怪波和暗怪波[54]

    Fig. 2.  The graphical evolution of rogue waves for the magnetization ${{m}} = \left( {{m_1}, {m_2}, {m_3}} \right)$ in Eq. (3) and (7), i.e., bright rogue waves (a)−(c) and dark rogue waves (d)−(f). The parameters are as follows: ${A_{\rm{s}}} = \sqrt 3 /2, {k_{\rm{s}}} = 1.5, \nu = {k_{\rm{s}}}\sqrt {1 - A_{\rm{s}}^2} $, and $\mu = \pm 0.75 \sqrt 3 $ with the sign $ \pm $ corresponding to the bright and dark rogue waves, respectively[54].

    图 3  方程(3)在As = 1, $\mu \to {A_{\rm{s}}}{k_{\rm{s}}}, \nu = {k_{\rm{s}}}\sqrt {1 - A_{\rm{s}}^2} $的条件下, 磁化分量${{m_3}}$四片花瓣结构的演化图. 参数如下: As = 1, $ {k_{\rm{s}}} = 0.9, \mu = 0.8999$

    Fig. 3.  The formation of magnetic petal in the component ${{m_3}}$ of Eq.(3) under the special condition of As = 1, $ \mu \to {A_{\rm{s}}}{k_{\rm{s}}}, \nu = {k_{\rm{s}}}\sqrt {1 - A_{\rm{s}}^2}$. The parameters are as follows: ${A_{\rm{s}}} = 1, {k_{\rm{s}}} = 0.9, \mu = 0.8999$.

    图 4  不同的参数${\mu _1}$下的磁振子密度分布图[53], 范围从0.09到0.29间隔0.05. 插图为怪波形成时的磁振子密度. 其余参数为${A_{\rm{c}}} = 0.2, {A_{\rm{J}}} = {k_c} = 0.1$

    Fig. 4.  The magnon density distribution against the background for the different parameter ${\mu _1}$, which ranges from 0.09 to 0.29 in 0.05 steps[53]. The inset figure is the magnon density distribution against the background for the excited formation of magnetic rogue wave. Other parameters are ${A_{\rm{c}}} = 0.2, {A_{\rm{J}}} = {k_{\rm{c}}} = 0.1$.

    图 5  (a)−(e)不同电流值激发怪波在区域$\left( {x, t} \right)$的分布图[53], 电流从0到0.8, 间隔为0.2; (f)不同电流怪波形成时的图形, 插图为磁振子积聚的最大时情况. 临界电流${A_{\rm{J}}} = 2{k_{\rm{c}}}$, 其它参数${A_{\rm{c}}} = {k_{\rm{c}}} = 0.2$

    Fig. 5.  (a)−(e) The formation region in space $\left( {x, t} \right)$ for magnetic rogue wave with different current[53]. The parameter ${A_J}$ ranges form 0 to 0.8 in 0.2 steps; (f) The nonuniform exchange of magnons between rogue wave and background for the different spin current. The inset figure in (f) denotes the maximal accumulation (or dissipation) process for the critical current value${A_{\rm{J}}} = 2{k_{\rm{c}}}$. Other parameters are ${A_{\rm{c}}} = {k_{\rm{c}}} = 0.2$.

  • [1]

    Bu K M, Kwon H Y, Kang S P, Kim H J, Won C 2013 J. Magn. Magn. Mater. 343 32Google Scholar

    [2]

    Moon J H, Seo S M, Lee K J, Kim K W, Ryu J, Lee H W, McMichael R D, Stiles M D 2013 Phys. Rev. B 88 184404Google Scholar

    [3]

    Yu X Z, Kanazawa N, Onose Y, Kimoto K, Zhang W Z, Ishiwata S, Matsui Y, Tokura Y 2011 Nat. Mater. 10 106Google Scholar

    [4]

    Rohart S, Thiaville A 2013 Phys. Rev. B 88 184422Google Scholar

    [5]

    Robler U K, Bogdanov A N, Pfleiderer C 2006 Nature 442 797Google Scholar

    [6]

    Allwood D A, Xiong G, Faulkner C C, Atkinson D, Petit D, Cowburn R P 2005 Science 309 1688Google Scholar

    [7]

    Parkin S S P, Hayashi M, Thomas L 2008 Science 320 190Google Scholar

    [8]

    Li Z D, Hu Y C, He P B, Sun L L 2018 Chin. Phys. B 27 077505Google Scholar

    [9]

    He P B, Xie X C, Liu W M 2005 Phys. Rev. B 72 172411Google Scholar

    [10]

    Tretiakov O A, Liu Y, Abanov Ar 2012 Phys. Rev. Lett. 108 247201Google Scholar

    [11]

    Li Q Y, Zhao F, He P B, Li Z D 2015 Chin. Phys. B 24 037508Google Scholar

    [12]

    Kosevich A M, Ivanov B A, Kovalev A S 1990 Phys. Rep. 194 117Google Scholar

    [13]

    Mikeska H J, Steiner M 1991 Adv. Phys. 40 191Google Scholar

    [14]

    Haazen P P J, Mure E, Franken J H, Lavrijsen R, H. Swagten J M, Koopmans B 2013 Nat. Mater. 12 299Google Scholar

    [15]

    Miron I M, Gaudin G, Auffret S, Rodmacq B, Schuhl A, Pizzini S, Vogel J, Gambardella P 2010 Nat. Mater. 9 230Google Scholar

    [16]

    Di K, Zhang V L, Lim H S, Ng S C, Kuok M H, Yu J, Yoon J, Qiu X, Yang H 2015 Phys. Rev. Lett. 114 047201Google Scholar

    [17]

    Wang W W, Albert M, Beg M, Bisotti M A, Chernyshenko D 2015 Phys. Rev. Lett. 114 087203Google Scholar

    [18]

    Li Z D, Cui H, Li Q Y, He P B 2018 Ann. Phys. 388 390Google Scholar

    [19]

    Chesi S, Coish W A 2015 Phys. Rev. B 91 245306Google Scholar

    [20]

    Li Z D, Liu F, Li Q Y, He P B 2015 J. Appl. Phys. 117 173906Google Scholar

    [21]

    Pfleiderer C, Julian S R, Lonzarich G G 2001 Nature 414 427Google Scholar

    [22]

    Uchida M, Onose Y, Matsui Y, Tokura Y 2006 Science 311 359Google Scholar

    [23]

    Meckler S, Mikuszeit N, Preßler A, Vedmedenko E Y, Pietzsch O, Wiesendanger R 2009 Phys. Rev. Lett. 103 157201Google Scholar

    [24]

    Moriya T 1960 Phys.Rev. 120 91Google Scholar

    [25]

    Zakharov D V, Deisenhofer J 2006 Phys. Rev. B 73 094452Google Scholar

    [26]

    Gangadharaiah S, Sun J M, Starykh O A 2008 Phys. Rev. B 78 054436Google Scholar

    [27]

    Albert F J, Emley N C, Myers E B, Ralph D C, Buhrman R A 2002 Phys. Rev. Lett. 89 226802Google Scholar

    [28]

    Mucciolo E R, Chamon C, Marcus C M 2002 Phys. Rev. Lett. 89 146802Google Scholar

    [29]

    Beach G S D, Knutson C, Nistor C, Tsoi M, Erskine J L 2006 Phys. Rev. Lett. 97 057203Google Scholar

    [30]

    Bertotti G, Serpico C, Mayergoyz I D, Magni A, Aquino M, Bonin R 2005 Phys. Rev. Lett. 94 127206Google Scholar

    [31]

    Garcia-Sanchez F, Borys P, Soucaille R, Adam J P, Stamps R L, Kim J V 2015 Phys. Rev. Lett. 114 247206Google Scholar

    [32]

    Katine J A, Albert F J, Buhrman R A, Myers E B, Ralph D C 2000 Phys. Rev. Lett. 84 3149Google Scholar

    [33]

    Tsoi M, Jansen A G M, Bass J, Chiang W C, Seck M, Tsoi V, Wyder P 1998 Phys. Rev. Lett. 80 4281Google Scholar

    [34]

    He P B, Liu W M 2005 Phys. Rev. B 72 064410Google Scholar

    [35]

    Li Z D, He P B, Liu W M 2014 Chin. Phys. B 23 117502Google Scholar

    [36]

    Tsoi M, Tsoi V, Bass J, Jansen A G M, Wyder P 2002 Phys. Rev. Lett. 89 246803Google Scholar

    [37]

    Tserkovnyak Y, Brataas A, Bauer G E W 2002 Phys. Rev. Lett. 88 117601Google Scholar

    [38]

    Li Z D, Li Q Y, He P B, Liang J Q, Liu W M, Fu G S 2010 Phys. Rev. A 81 015602Google Scholar

    [39]

    Freimuth F, Blugel S, Mokrousov Y 2014 Phys. Rev. B 90 174423Google Scholar

    [40]

    Santos T S, Lee J S, Migdal P, Lekshmi I C, Satpati B, Moodera J S 2007 Phys. Rev. Lett. 98 016601Google Scholar

    [41]

    Li Z, Zhang S 2004 Phys. Rev. Lett. 92 207203Google Scholar

    [42]

    Kasai S, Nakatani Y, Kobayashi K, Kohno H, Ono T 2006 Phys. Rev. Lett. 97 107204Google Scholar

    [43]

    Kasai S, Fischer P, Im M Y, Yamada K, Nakatani Y, Kobayashi K, Kohno H, Ono T 2008 Phys. Rev. Lett. 101 237203; Sugimoto S, Fukuma Y, Otani Y C 2011 IEEI. T. Magn. 47 2951

    [44]

    Yamada K, Kasai S, Nakatani Y, Kobayashi K, Kohno H, Thiaville A, Ono T 2007 Nat. Mater. 6 270; Moriya R, Thomas L, Hayashi M, Bazaliy Y B, Rettner C, Parkin S S P 2008 Nat. Phys. 4 368

    [45]

    Zhao L C, Ling L 2016 J. Opt. Soc. Am. B 33 850Google Scholar

    [46]

    Liu C, Yang Z Y, Zhao L C, Yang W L 2015 Phys. Rev. E 91 022904Google Scholar

    [47]

    Duan L, Zhao L C, Xu W H 2017 Phys. Rev. E 95 042212Google Scholar

    [48]

    Wang L, X Wuan, Zhang H Y 2018 Phys. Lett. A 382 2650Google Scholar

    [49]

    Wang L, Liu C, Wu X, Wang X, Sun W R 2018 Nonlinear Dyn. 94 977Google Scholar

    [50]

    Zakharov V E, Gelash A A 2013 Phys. Rev. Lett. 111 054101Google Scholar

    [51]

    Gelash A A, Zakharov V E 2014 Nonlinearity. 27 R1Google Scholar

    [52]

    Yan P, Wang X S, Wang X R 2011 Phys. Rev. Lett. 107 177207Google Scholar

    [53]

    Zhao F, Li Z D, Li Q Y, Wen L, Fu G S, Liu W M 2012 Ann. Phys. 327 2085Google Scholar

    [54]

    Li Z D, Li Q Y, Xu T F, He P B 2016 Phys. Rev. E 94 042220Google Scholar

    [55]

    Li Q Y, Li Z D, He P B, Song W W, Fu G S 2010 Can. J. Phys. 88 9

    [56]

    Hasegawa A 1984 Opt. Lett. 9 288; Tai K, Tomita A, Jewell J L, Hasegawa A 1986 Appl. Phys. Lett. 49 236

    [57]

    Akhmediev N N 2001 Nature. 413 267; Van Simaeys G, Emplit P, Haelterman M 2001 Phys. Rev. Lett. 87 033902; Mussot A, Kudlinski A, Droques M, Szriftgiser P, Akhmediev N 2014 Phys. Rev. X 4 011054

    [58]

    Zhang H Q, Tian B, Xing L, Meng X H 2010 Physica A 389 367Google Scholar

    [59]

    Matveev V B, Salli M A 1991 Darboux Transformations and Solitons, Vol. 5 (Berlin: Springer) pp7−15

    [60]

    谷超豪, 胡和生, 周子翔 2005 孤立子理论中的达布变换及其几何应用(第二版)(上海: 上海科学技术出版社)第18−24页

    Gu C H, Hu H S, Zhou Z X 2005 Darboux Transformation in Soliton Theory and Its Geometric Applications (Shanghai: Scientific and Technical Publishers) pp18−24 (in Chinese).

    [61]

    Li Z D, Wu X, Li Q Y, He P B 2016 Chin. Phys. B 25 010507Google Scholar

    [62]

    Li Z D, Huo C Z, Li Q Y, He P B, Xu T F 2018 Chin. Phys. B 27 040505Google Scholar

    [63]

    Ho T L 1998 Phys. Rev. Lett. 81 742; Law C K, Ohmi T, Machida K, 1998 J. Phys. Soc. Jpn. 67 1822; Law C K, Pu H, Bigelow N P 1998 Phys. Rev. Lett. 81 5257

    [64]

    Pu H, Zhang W P, Meystre P 2001 Phys. Rev. Lett. 87 140405Google Scholar

    [65]

    Li Z D, He P B, Li L, Liang J Q, Liu W M 2005 Phys. Rev. A 71 053611Google Scholar

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出版历程
  • 收稿日期:  2019-09-06
  • 修回日期:  2019-12-02
  • 上网日期:  2019-12-17
  • 刊出日期:  2020-01-05

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