基本非线性波与调制不稳定性的精确对应

段亮; 刘冲; 赵立臣; 杨战营

doi:10.7498/aps.69.20191385

摘要

非线性波作为非线性动力学研究中的重要课题之一, 普遍存在于各种复杂物理系统中. 理解非线性波的产生机制、确定它们的激发条件对于非线性波的实验实现、动力学特征的探测和应用是至关重要的. 本文简要综述了近年来非线性波的实验和理论研究进展, 回顾了非线性波的产生机制. 基于非线性可积模型中的严格解和线性稳定分析结果, 系统讨论了如何建立基本非线性波与调制不稳定性的精确对应关系. 详细介绍了近来发现的扰动能量和相对相位在确定非线性波激发条件中的重要作用, 并提议了一组能够确定非线性波激发条件的完备参数. 基于完备的激发参数, 给出了多种基本非线性波的激发条件和相图. 这些结果有望用于实现多种局域波的可控激发, 并可以推广到更多非线性系统中的激发相图研究.

关键词:

Abstract

Nonlinear waves are ubiquitous in various physical systems, and they have become one of the research hotspots in nonlinear physics. For the experimental realization, observation and application of nonlinear waves, it is very important to understand the generation mechanism, and determine the essential excitation conditions of various nonlinear waves. In this paper, we first briefly review the experimental and theoretical research progress of nonlinear waves in recent years. Based on the exact nonlinear wave solutions and linear stability analysis results, we systemically discuss how to establish the quantitative relations between fundamental nonlinear waves and modulation instability. These relations would deepen our understanding on the mechanism of nonlinear waves. To solve the excitation conditions degenerations problem for some nonlinear waves, we further introduce the perturbation energy and relative phase to determine the excitation conditions of nonlinear waves. Finally, we present a set of complete parameters that can determine the excitation conditions of nonlinear waves, and give the excitation conditions and phase diagrams of the fundamental nonlinear waves. These results can be used to realize controllable excitation of nonlinear waves, and could be extended to many other nonlinear systems.

Keywords:

作者及机构信息

1.
西北大学物理学院, 西安　710127

2.
陕西省理论物理前沿重点实验室, 西安　710069

通信作者: 杨战营, zyyang@nwu.edu.cn

基金项目: 国家自然科学基金(批准号: 11875220, 11775176)

Authors and contacts

1.
School of Physics, Northwest University, Xi’an 710127, China

2.
Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi’an 710069, China

Corresponding author: Yang Zhan-Ying, zyyang@nwu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11875220, 11775176)

文章全文 : translate this paragraph

参考文献

[1]	Guo B, Pan X 1990 Chin. Phys. Lett. 7 241 Google Scholar
[2]	Lou S Y, Ni G J, Huang G X 1992 Commun. Theor. Phys. 17 67 Google Scholar
[3]	Chabchoub A, Hoffmann N P, Akhmediev N 2011 Phys. Rev. Lett. 106 204502 Google Scholar
[4]	Chabchoub A, Hoffmann N, Onorato M, et al. 2012 Phys. Rev. X 2 011015
[5]	Chabchoub A, Hoffmann N, Onorato M, et al. 2012 Phys. Rev. E 86 056601 Google Scholar
[6]	Chen Z, Segev M, et al 1996 Opt. Lett. 21 1821 Google Scholar
[7]	Chen Z, Segev M, et al 1997 J. Opt. Soc. Am. B 14 1407 Google Scholar
[8]	Guo Q, Luo B, Yi F, Chi S, Xie Y 2004 Phys. Rev. E 69 016602 Google Scholar
[9]	Deng D, Guo Q 2007 Opt. Lett. 32 3206 Google Scholar
[10]	Solli D R, Ropers C, Koonath P, et al. 2007 Nature 450 1054 Google Scholar
[11]	Kibler B, Fatome J, Finot C, et al. 2010 Nat. Phys. 6 790 Google Scholar
[12]	Dudley J M, Genty G, Dias F, et al. 2009 Opt. Express 17 21497 Google Scholar
[13]	Liu X 2011 Phys. Rev. A 84 053828 Google Scholar
[14]	Kibler B, Fatome J, Finot C, et al. 2012 Sci. Rep. 2 463 Google Scholar
[15]	Jia J, Lin J 2012 Opt. Express 20 7469 Google Scholar
[16]	Zhang Y, Belic M, Wu Z, Zheng H, Lu K, Li Y, Zhang Y 2013 Opt. Lett. 38 4585 Google Scholar
[17]	Lin J, Chen W W, Jia J 2014 J. Opt. Soc. Am. A 31 188 Google Scholar
[18]	Liu W, Pang L, Han H, Shen Z, Lei M, Teng H, Wei Z 2016 Photon. Research 4 111 Google Scholar
[19]	Liu W, Pang L, Yan H, Ma G, Lei M, Wei Z 2016 EuroPhys. Lett. 116 64002 Google Scholar
[20]	Liu X, Yao X, Cui Y 2018 Phys. Rev. Lett. 121 023905 Google Scholar
[21]	Liu X, Popa D, Akhmediev N 2019 Phys. Rev. Lett. 123 093901 Google Scholar
[22]	Bailung H, Sharma S K, Nakamura Y 2011 Phys. Rev. Lett. 107 255005 Google Scholar
[23]	Tsai Y Y, Tsai J Y, Lin I 2016 Nat. Phys. 12 573 Google Scholar
[24]	Zhang W, Walls D F 1994 Phys. Rev. Lett. 72 60 Google Scholar
[25]	Zhang W, Walls D F 1994 Phys. Rev. A 49 3799 Google Scholar
[26]	Burger S, Bongs K, Dettmer S, et al. 1999 Phys. Rev. Lett. 83 5198 Google Scholar
[27]	Denschlag J, Simsarian J E, Feder D L, et al. 2000 Science 287 97 Google Scholar
[28]	Huang G X 2001 Chin. Phys. Lett. 18 628 Google Scholar
[29]	Khaykovich L, Schreck F, Ferrari G, et al. 2002 Science 296 1290 Google Scholar
[30]	Strecker K E, Partridge G B, Truscott A G, et al. 2002 Nature 417 150 Google Scholar
[31]	Liang Z X, Zhang Z D, Liu W M 2005 Phys. Rev. Lett. 94 050402 Google Scholar
[32]	Bludov Y V, Konotop V V, Akhmediev N 2009 Phys. Rev. A 80 033610 Google Scholar
[33]	Zhao D, Luo H G, Chai H Y 2008 Phys. Lett. A 372 5644 Google Scholar
[34]	Feng B, Zhao D 2016 J. Differ. Equations 260 2973 Google Scholar
[35]	Zeng J, Malomed B A 2017 Phys. Rev. E 95 052214 Google Scholar
[36]	Yao Y Q, Han W, Li J, Liu W M 2018 J. Phys. B 51 105001 Google Scholar
[37]	Wang D S, Liu J, Wang L 2018 Phys. Lett. A 382 799 Google Scholar
[38]	He Z M, Wen L, Wang Y J, Chen G P, Tan R B, Dai C Q, Zhang X F 2019 Phys. Rev. E 99 062216 Google Scholar
[39]	Shats M, Punzmann H, Xia H 2010 Phys. Rev. Lett. 104 104503 Google Scholar
[40]	Daniel M, Kavitha L, Amuda R 1999 Phys. Rev. B 59 13774 Google Scholar
[41]	Daniel M, Beula J 2009 Chaos, Solitons Fractals 41 1842
[42]	Daniel M, Beula J 2009 Phys. Lett. A 373 2841 Google Scholar
[43]	Zhao F, Li Z D, Li Q Y, et al 2012 Ann. Phys. 327 2085 Google Scholar
[44]	Qi J W, Li Z D, Yang Z Y, et al. 2017 Phys. Lett. A 381 1874 Google Scholar
[45]	Yan Z 2010 Commun. Theor. Phys. 54 947 Google Scholar
[46]	Yan Z 2011 Phys. Lett. A 375 4274 Google Scholar
[47]	Wu Y, Zhao L C, Lei X K 2015 Eur. Phys. J. B 88 297 Google Scholar
[48]	Zheludev N I, Kivshar Y S 2012 Nat. Mater. 11 917 Google Scholar
[49]	Wen S, Wang Y, Su W, Xiang Y, Fu X, Fan D 2006 Phys. Rev. E 73 036617 Google Scholar
[50]	Xiong H, Gan J, Wu Y 2017 Phys. Rev. Lett. 119 153901 Google Scholar
[51]	Konotop V V, Yang J, Zezyulin D A 2016 Rev. Mod. Phys. 88 035002 Google Scholar
[52]	Lou S Y, Huang F 2017 Sci. Rep. 7 869 Google Scholar
[53]	Polo J, Ahufinger V 2013 Phys. Rev. A 88 053628 Google Scholar
[54]	McDonald G D, Kuhn C C N, Hardman K S, et al. 2014 Phys. Rev. Lett. 113 013002 Google Scholar
[55]	Helm J L, Cornish S L, Gardiner S A 2015 Phys. Rev. Lett. 114 134101 Google Scholar
[56]	Zhao L C, Ling L, Yang Z Y, Liu J 2016 Nonlinear Dyn. 83 659 Google Scholar
[57]	Solli D R, Ropers C, Jalali B 2008 Phys. Rev. Lett. 101 233902 Google Scholar
[58]	Frisquet B, Chabchoub A, Fatome J, et al. 2014 Phys. Rev. A 89 023821 Google Scholar
[59]	Gertjerenken B, Billam T P, Blackley C L, et al. 2013 Phys. Rev. Lett. 111 100406 Google Scholar
[60]	Fatome J, Kibler B, Finot C 2013 Opt. Lett. 38 1663 Google Scholar
[61]	Yang G, Wang Y, Qin Z, et al. 2014 Phys. Rev. E 90 062909 Google Scholar
[62]	Zhao L C 2018 Phys. Rev. E 97 062201 Google Scholar
[63]	Hasegawa A, Tappert F 1973 Appl. Phys. Lett. 23 171 Google Scholar
[64]	Zakharov V E, Shabat A B 1973 Sov. Phys. JETP 37 823
[65]	Kivshar Y S, Luther-Davies B 1998 Phys. Rep. 298 81 Google Scholar
[66]	Kivshar Y S 1991 Phys. Rev. A 43 1677 Google Scholar
[67]	Kivshar Y S, Afanasjev V V 1991 Phys. Rev. A 44 R1446 Google Scholar
[68]	Dong G, Liu Z 1996 Opt. Commun. 128 8 Google Scholar
[69]	Li Z, Li L, Tian H, et al. 2000 Phys. Rev. Lett. 84 4096 Google Scholar
[70]	Zhao L C, Li S C, Ling L 2014 Phys. Rev. E 89 023210 Google Scholar
[71]	Liu C, Yang Z Y, Zhao L C, et al. 2015 Phys. Rev. E 91 022904 Google Scholar
[72]	Ren Y, Yang Z Y, Liu C, et al. 2015 Phys. Lett. A 379 2991 Google Scholar
[73]	Liu C, Yang Z Y, Zhao L C, et al. 2016 Phys. Rev. E 94 042221 Google Scholar
[74]	Zhao L C, Li S C, Ling L 2016 Phys. Rev. E 93 032215 Google Scholar
[75]	Peregrine D H 1983 J. Australas. Math. Soc. Ser. B 25 16 Google Scholar
[76]	Akhmediev N N, Korneev V I 1986 Theor. Math. Phys. 1089
[77]	Kuznetsov E A 1977 Akademiia Nauk SSSR Doklady 236 575
[78]	Ma Y C 1979 Stud. Appl. Math. 60 43 Google Scholar
[79]	Tajiri M, Watanabe Y 1998 Phys. Rev. E 57 3510 Google Scholar
[80]	Priya N V, Senthilvelan M, Lakshmanan M 2013 Phys. Rev. E 88 022918 Google Scholar
[81]	刘冲 2016 博士学位论文(西安: 西北大学) Liu C 2016 Ph. D. Dissertation (Xi’an: Northwest University) (in Chinese)
[82]	Chowdury A, Ankiewicz A, Akhmediev N 2015 Proc. R. Soc. A 471 20150130 Google Scholar
[83]	Zakharov V E, Gelash A A 2013 Phys. Rev. Lett. 111 054101 Google Scholar
[84]	Kibler B, Chabchoub A, Gelash A, Akhmediev N, Zakharov V E 2015 Phys. Rev. X 5 041026
[85]	Zhang J H, Wang L, Liu C 2017 Proc. R. Soc. A 473 20160681 Google Scholar
[86]	Liu C, Ren Y, Yang Z Y, Yang W L 2017 Chaos 27 083120 Google Scholar
[87]	Liu C, Yang Z Y, Yang W L 2018 Chaos 28 083110 Google Scholar
[88]	Ren Y, Liu C, Yang Z Y, Yang W L 2018 Phys. Rev. E 98 062223 Google Scholar
[89]	Ren Y, Wang X, Liu C, Yang Z Y, Yang W L 2018 Commun. Nonlinear Sci. Numer. Simul. 63 161 Google Scholar
[90]	Xu G, Gelash A, Chabchoub A, Zakharov V, Kibler B 2019 Phys. Rev. Lett. 122 084101 Google Scholar
[91]	Liu C, Yang Z Y, Yang W L, Akhmediev N 2019 J. Opt. Soc. Am. B 36 1294 Google Scholar
[92]	Guo B, Ling L 2011 Chin. Phys. Lett. 28 110202 Google Scholar
[93]	Guo B, Ling L, Liu Q P 2012 Phys. Rev. E 85 026607 Google Scholar
[94]	Ling L, Zhao L C 2013 Phys. Rev. E 88 043201
[95]	Ling L, Guo B, Zhao L C 2014 Phys. Rev. E 89 041201(R)
[96]	Zhao L C, Liu J 2012 J. Opt. Soc. Am. B 29 3119 Google Scholar
[97]	Zhao L C, Liu J 2013 Phys. Rev. E 87 013201 Google Scholar
[98]	Zhao L C, Xin G G, Yang Z Y 2014 Phys. Rev. E 90 022918 Google Scholar
[99]	Baronio F, Conforti M, Degasperis A, et al. 2013 Phys. Rev. Lett. 111 114101 Google Scholar
[100]	Baronio F, Conforti M, Degasperis A, et al. 2014 Phys. Rev. Lett. 113 034101 Google Scholar
[101]	Zhao L C, Ling L, Yang Z Y, Liu J 2015 Commun. Nonlinear Sci. Numer. Simul. 23 21 Google Scholar
[102]	Chen S, Cai X M, Grelu P, et al. 2016 Opt. Express 24 5886 Google Scholar
[103]	Ling L, Zhao L C, Yang Z Y, et al. 2017 Phys. Rev. E 96 022211 Google Scholar
[104]	Yan Z 2015 Nonlinear Dyn. 79 2515 Google Scholar
[105]	Wen X Y, Yang Y, Yan Z 2015 Phys. Rev. E 92 012917 Google Scholar
[106]	Zhao L C, Guo B, Ling L 2016 J. Math. Phys. 57 043508 Google Scholar
[107]	Zhang G, Yan Z, Wen X Y, Chen Y 2017 Phys. Rev. E 95 042201 Google Scholar
[108]	Li M, Shui J J, Xu T 2018 Appl. Math. Lett. 83 110 Google Scholar
[109]	Zhao L C, Ling L 2014 arXiv: 1410.7536; 2016 J. Opt. Soc. Am. B 33 850
[110]	Baronio F, Chen S, Grelu P, et al. 2015 Phys. Rev. A 91 033804 Google Scholar
[111]	He J S, Xu S, Porsezian K 2012 J. Phys. Soc. Jpn. 81 124007 Google Scholar
[112]	Xu S, He J S, Cheng Y, Porseizan K 2015 Math. Meth. Appl. Sci. 38 1106?Google Scholar
[113]	Xu S, He J S 2012 J. Math. Phys. 53 063507 Google Scholar
[114]	Zhao L C, Yang Z Y, Ling L 2014 J. Phys. Soc. Jpn. 83 104401 Google Scholar
[115]	Zhao L C, Liu C, Yang Z Y 2015 Commun. Nonlinear Sci. Numer. Simul. 20 9 Google Scholar
[116]	Akhmediev N, Soto-Crespo J M, Ankiewicz A 2009 Phys. Rev. A 80 043818 Google Scholar
[117]	He J S, Zhang H R, Wang L H, Porsezian K, Fokas A S 2013 Phys. Rev. E 87 052914 Google Scholar
[118]	Wang L, He J S, Xu H, Wang J, Porsezian K 2017 Phys. Rev. E 95 042217 Google Scholar
[119]	Zakharov V E, Ostrovsky L A 2009 Physica D 238 540 Google Scholar
[120]	Hammani K, Wetzel B, Kibler B, et al. 2011 Opt. Lett. 36 2140 Google Scholar
[121]	Soto-Crespo J M, Ankiewicz A, Devine N, et al. 2012 J. Opt. Soc. Am. B 29 1930 Google Scholar
[122]	Forest M G, McLaughlin D W, Muraki D J, et al. 2000 J. Nonlinear Sci. 10 291 Google Scholar
[123]	Hirota R 1973 J. Math. Phys. 14 805 Google Scholar
[124]	Sasa N, Satsuma J 1991 J. Phys. Soc. Jpn. 60 409 Google Scholar
[125]	Duan L, Zhao L C, Xu W H, et al. 2017 Phys. Rev. E 95 042212 Google Scholar
[126]	Li P, Wang L, Kong L Q, et al. 2018 Appl. Math. Lett. 85 110 Google Scholar
[127]	Erkintalo M, Genty G, Wetzel B, et al. 2011 Phys. Lett. A 375 2029 Google Scholar
[128]	Agrawal G P 2007 Nonlinear Fiber Optics (Massachusetts: Academic Press)
[129]	Kevrekidis P G, Frantzeskakis D, Carretero-Gonzalez R 2007 Emergent Nonlinear Phenomena in BoseEinstein Condensates: Theory and Experiment (New York: Springer Science and Business Media)
[130]	Kawaguchi Y, Ueda M 2012 Phys. Rep. 520 253 Google Scholar
[131]	Gao P, Duan L, Zhao L C, et al. 2019 Chaos 29 083112 Google Scholar
[132]	Duan L, Yang Z Y, Zhao L C, et al. 2016 J. Mod. Opt. 63 1397 Google Scholar
[133]	Liu X S, Zhao L C, Duan L, et al. 2017 Chin. Phys. B 26 120503 Google Scholar
[134]	Liu C, Yang Z Y, Zhao L C, et al. 2015 Ann. Phys. 362 130 Google Scholar
[135]	Wang L, Li S, Qi F H 2016 Nonlinear Dyn. 85 389 Google Scholar
[136]	Wang X, Liu C, Wang L 2017 Chaos 27 093106 Google Scholar
[137]	Blanco-Redondo A, De Sterke C M, Sipe J E, et al. 2016 Nat. Commun. 7 10427 Google Scholar
[138]	Daniel M, Kavitha L 2001 Phys. Rev. B 63 172302 Google Scholar
[139]	Tao Y, He J S 2012 Phys. Rev. E 85 026601 Google Scholar
[140]	Wang L H, Porsezian K, He J S 2013 Phys. Rev. E 87 053202 Google Scholar
[141]	Ankiewicz A, Wang Y, Wabnitz S, et al. 2014 Phys. Rev. E 89 012907 Google Scholar
[142]	Chowdury A, Kedziora D J, Ankiewicz A, et al. 2014 Phys. Rev. E 90 032922 Google Scholar
[143]	Yang Y, Yan Z, Malomed B A 2015 Chaos 25 103112 Google Scholar
[144]	Chowdury A, Kedziora D J, Ankiewicz A, et al. 2015 Phys. Rev. E 91 032928 Google Scholar
[145]	Ankiewicz A, Kedziora D J, Chowdury A, et al. 2016 Phys. Rev. E 93 012206 Google Scholar
[146]	Wang L, Zhang J H, Wang Z Q, et al. 2016 Phys. Rev. E 93 012214 Google Scholar
[147]	Zhao L C, Ling L, Yang Z Y 2018 Phys. Rev. E 97 022218 Google Scholar
[148]	Duan L, Yang Z Y, Gao P, et al. 2019 Phys. Rev. E 99 012216 Google Scholar
[149]	Wen X Y, Yan Z, Boris A, Malomed 216 Chaos 26 123110
[150]	Wen X Y, Yan Z 2018 J. Math. Phys. 59 073511 Google Scholar
[151]	Wen X Y, Yan Z 2017 Commun. Nonlinear. Sci. Numer. Simul. 43 311 Google Scholar
[152]	Ohta Y, Yang J 2012 Phys. Rev. E 86 036604 Google Scholar
[153]	Yang B, Chen Y 2018 Appl. Math. Lett. 82 43 Google Scholar
[154]	Li M, Xu T, Meng D 2016 J. Phys. Soc. Jpn. 85 124001 Google Scholar
[155]	Xu T, Li H, Zhang H, et al. 2017 Appl. Math. Lett. 63 88 Google Scholar
[156]	Dai C Q, Zhu H P 2013 J. Opt. Soc. Am. B 30 3291 Google Scholar
[157]	Chen J, Chen Y, Feng B F, Maruno K 2015 Phys. Lett. A 379 1510 Google Scholar
[158]	Zhang X, Chen Y, Tang X 2018 Comput. Math. Appl. 76 1938 Google Scholar
[159]	Qian C, Rao J G, Liu Y B, He J S 2016 Chin. Phys. Lett. 33 110201 Google Scholar
[160]	Zhang X, Chen Y 2017 Commun. Nonlinear. Sci. Numer. Simul. 52 24 Google Scholar
[161]	Liu Y K, and Li B 2017 Chin. Phys. Lett. 34 010202 Google Scholar
[162]	Charalampidis E G, Wang W, Kevrekidis P G, Frantzeskakis D J, Cuevas-Maraver J 2016 Phys. Rev. A 93 063623 Google Scholar
[163]	Wang W, Kevrekidis P G, Carretero-Gonzalez R, Frantzeskakis D J 2016 Phys. Rev. A 93 023630 Google Scholar
[164]	Rao J, Porsezian K, He J S 2017 Chaos 27 083115??Google Scholar
[165]	Rao J, Porsezian K, He J S, Kanna T 2018 Proc. R. Soc. A 474 20170627
[166]	Zeng L, Zeng J, Kartashov Y V, Malomed B A 2019 Opt. Lett. 44 1206 Google Scholar

施引文献

图 1 自散焦的两组分耦合非线性薛定谔系统的调制不稳定增益的分布　(a)调制不稳定增益在$(\varOmega, \omega)$平面的分布, 绿色点状曲线表示调制不稳定区的边界; (b)调制不稳定性在$(\varOmega, a_{1})$平面的分布

Fig. 1. Modulation instability distributions of the defocusing two component coupled nonlinear Schrödinger system: (a) Modulation instability distribution in the $(\varOmega, \omega)$ plane, green dot curves are the boundary of the modulation instability regime; (b) modulation instability distribution in the $(\varOmega, a_{1})$ plane.

下载: 全尺寸图片幻灯片

图 2 标准非线性薛定谔系统的调制不稳定增益分布和基本非线性波激发的相图　(a1)和(b1)分别为调制不稳定增益在$(\omega, \varOmega)$平面和$(a, \varOmega)$平面的分布. “MI”和“MS”分别表示调制不稳定性和调制稳定性, 红色虚线是共振线; (a2)和(b2)分别为基本非线性波在(a1)和(b1)中调制不稳定增益分布平面的相图. “AB”,“RW”和“KM”分别为Akhmediev呼吸子、怪波和Kuznetsov-Ma呼吸子

Fig. 2. Modulation instability distributions and phase diagrams of fundamental nonlinear waves in standard nonlinear Schrödinger system: (a1) and (b1) are the distributions of the modulation instability gain in the $(\omega, \varOmega)$ plane and the $(a, \varOmega)$, respectively. “MI” and “MS” denote modulation instability and modulation stability, respectively. the red dotted line is the resonance line; (a2) and (b2) are the phase diagrams of fundamental nonlinear waves on the modulation instability gain distribution planes correspond to (a1) and (b1), respectively. "AB", "RW" and "KM" denote Akhmediev breather, rogue wave and Kuznetsov-Ma breather, respectively.

下载: 全尺寸图片幻灯片

图 3 Sasa-Satsuma系统的调制不稳定增益分布和基本非线性波激发的相图　(a) Sasa-Satsuma系统中调制不稳定增益在背景频率$\omega$和扰动频率$\varOmega$平面的分布. “MI”和“MS”分别表示调制不稳定和调制稳定, 黄颜色圆点为共振线上临界点; (b)非线性波在调制不稳定增益分布平面的相图. “AB”, “RW” 和“KM” 分别为Akhmediev呼吸子、怪波和Kuznetsov-Ma呼吸子; “WS”, “WST”, “AD”和Periodic wave分别表示W形孤子、W形孤子链、反暗孤子和周期波

Fig. 3. Modulation instability distributions and phase diagrams of fundamental nonlinear waves in Sasa-Satsuma system: (a) Distributions of the modulation instability gain in the background frequency $\omega$ and perturbation frequency$\varOmega$ plane. “MI” and “MS” denote modulation instability and modulation stability, respectively. The yellow dots are the critical points on the resonance line; (b) phase diagrams of nonlinear waves in the modulation instability gain distribution planes. “AB”, “RW” and “KM” denote Akhmediev breather, rogue wave and Kuznetsov-Ma breather, respectively; “WS”, “WST” and “AD” denote the W-shaped soliton, W-shaped soliton train and anti-dark soliton, respectively.

下载: 全尺寸图片幻灯片

图 4 Hirota系统中的调制不稳定增益分布和基本非线性波激发的相图　(a) Hirota系统中调制不稳定增益在背景频率$\omega$和扰动频率$\varOmega$平面的分布. “MI”和“MS”分别表示调制不稳定和调制稳定; (b)非线性波在调制不稳定增益分布平面的相图. “AB”, “RW” 和“KM” 分别为Akhmediev呼吸子、怪波和Kuznetsov-Ma呼吸子; “WS”, “AD”, “PW”和“MPS”分别表示W形孤子、反暗孤子、周期波和多峰孤子

Fig. 4. Modulation instability distributions and phase diagrams of fundamental nonlinear waves in Hirota system; (a) Distributions of the modulation instability gain in the background frequency $\omega$ and perturbation frequency$\varOmega$ plane. “MI” and “MS” denote modulation instability and modulation stability, respectively; (b) phase diagrams of nonlinear waves in the modulation instability gain distribution planes. “AB”, “RW” and “KM” denote Akhmediev breather, rogue wave and Kuznetsov-Ma breather, respectively; “WS”, “AD”, “PW” and “MPS” denote the W-shaped soliton, anti-dark soliton, periodic wave and multi-peak soliton, respectively.

下载: 全尺寸图片幻灯片

图 5 四阶非线性薛定谔系统调制不稳定增益分布和基本非线性波激发的相图　(a) 调制不稳定增益在背景频率$\omega$和扰动频率$\varOmega$平面的分布, “MI”和“MS” 分别表示调制不稳定性和调制稳定性; (b),(c) 基本非线性波在背景频率$\omega$和扰动频率$\varOmega$平面的相图, “AB”, “RW”, “KM”、“PW”, “WST”, “WS$_{\rm r}$”, “$\rm WS_{ {nr}}$” 和“AD”分别为Akhmediev呼吸子、怪波、Kuznetsov-Ma呼吸子、周期波、W形孤子链、有理的W形孤子、非有理的W形孤子和反暗孤子

Fig. 5. Modulation instability distributions and phase diagrams of fundamental nonlinear waves in fourth-order nonlinear Schrödinger system: (a) Distributions of the modulation instability gain in the background frequency $\omega$ and perturbation frequency$\varOmega$ plane. “MI” and “MS” denote modulation instability and modulation stability, respectively; (b), (c) phase diagrams of nonlinear waves in the background frequency $\omega$ and perturbation frequency $\varOmega$ plane. “AB”, “RW”, “KM”, “PW”, “WST”, “WS$_{\rm r}$”, “WS$\rm WS_{{nr}}$” and “AD” denote Akhmediev breather, rogue wave, Kuznetsov-Ma breather, periodic wave, W-shaped soliton train, rational W-shaped soliton, nonrational W-shaped soliton and anti-dark soliton, respectively.

下载: 全尺寸图片幻灯片

图 6 不同系统中平面波背景上基本非线性波在背景频率$\omega$, 扰动频率$\varOmega$, 扰动能量$\varepsilon$和相对相位$\varphi$空间的相图　(a) 四阶非线性薛定谔系统, 参数取$\beta={1}/{12}$, $\gamma=-{1}/{36}$, $a=1$; (b) Hirota系统, 参数取$\beta={1}/{12}$, $\gamma=0$, $a=1$; (c)非线性薛定谔系统, 参数取$\beta={1}/{12}$, $\gamma=0$, $a=1$; (d)反暗孤子和非有理W形孤子依赖于相对相位的相图; (e)周期波, W形孤子链和有理W形孤子在$(\varphi, \varOmega)$平面的相图. 图中“TW”, “KM”, “AB”, “RW”, “MPS”, “AD”, “WS$_{\rm {nr}}$”, “PW”, “WST”和“WS$_{\rm r}$”分别表示Tajiri-Watanabe呼吸子、Kuznetsov-Ma呼吸子、Akhmediev呼吸子、怪波、多峰孤子、反暗孤子、非有理W形孤子、周期波、W形孤子链和有理W形孤子

Fig. 6. Phase diagrams of nonlinear waves in the background frequency $\omega$, perturbation frequency $\varOmega$, perturbation energy $\varepsilon$ and relative phase $\varphi$ space for different systems: (a) Fourth-order nonlinear Schrödinger system. Parameters are $\beta={1}/{12}$, $\gamma=-{1}/{36}$, $a=1$; (b) hirota system. Parameters are $\beta={1}/{12}$, $\gamma=0$, $a=1$; (c) nonlinear Schrödinger system. Parameters are $\beta=\gamma=0$, $a=1$; (d) phase diagram of anti-dark soliton and nonrational W-shaped soliton in relative phase space; (e) phase diagram of periodic wave, W-shaped soliton train and rational W-shaped soliton in the $(\varphi, \varOmega)$ plane. “TW”, “KM”, “AB”, “RW”, “MPS”, “AD”, “WS$_{\rm {nr}}$”, “PW”, “WST” and “WS$_{\rm r}$” denote Tajiri-Watanabe breather, Kuznetsov-Ma breather, Akhmediev breather, rogue wave, multi-peak soliton, anti-dark soliton, nonrational W-shaped soliton, periodic wave, W-shaped soliton train and rational W-shaped soliton.

下载: 全尺寸图片幻灯片

图 7 不同非线性波的转换关系　(a) 呼吸子和怪波之间的转换关系； (b) 孤子和周期波之间的转换关系. 图中“TW”, “KM”, “AB”, “RW”分别为Tajiri-Watanabe呼吸子、Kuznetsov-Ma呼吸子、Akhmediev呼吸子和怪波, “MPS”, “AD”, “WS$_{\rm {nr}}$”, “PW”, “WST”和“WS$_{\rm r}$” 分别表示多峰孤子、反暗孤子、非有理W形孤子、周期波、W形孤子链和有理W形孤子

Fig. 7. Conversion relationship of different nonlinear waves: (a) Conversion relationship between breathers and rogue wave; (b) conversion relationship between the solitons and periodic waves. “TW”, “KM”, “AB”, “RW”, “MPS”, “AD”, “WS$_{\rm {nr}}$”, “PW”, “WST” and “WS$_{\rm r}$” denote Tajiri-Watanabe breather, Kuznetsov-Ma breather, Akhmediev breather, rogue wave, multi-peak soliton, anti-dark soliton, nonrational W-shaped soliton, periodic wave, W-shaped soliton train and rational W-shaped soliton.

下载: 全尺寸图片幻灯片

表 1 基本非线性波的激发条件

Table 1. Excitation conditions of fundamental nonlinear waves.

激发条件				非线性波类型
$\varOmega$	$\omega$	$\varepsilon$	$\varphi$	非线性波类型
0	$\omega^{2}-\alpha\neq0 $	0	$\varphi\in \left(\dfrac{{\text{π}}}{2}, \dfrac{3{\text{π}}}{2}\right)+2 n{\text{π}}$	怪波
0	$\omega^{2}-\alpha=0$, $\alpha\geqslant 0$	0		有理W形孤子
0	$\omega^{2}-\dfrac{\varepsilon^{2}}{24}-\alpha\neq0$, $\varepsilon>0$		$\varphi\in\mathbb{R}$	Kuznetsov-Ma呼吸子
	$\omega^{2}-\dfrac{\varepsilon^{2}}{24}-\alpha=0$, $\varepsilon>0$		$\varphi\in\left(\dfrac{{\text{π}}}{2},\right. \left.\dfrac{3{\text{π}}}{2}\right]+2 n{\text{π}}$	非有理W形孤子
	$\omega^{2}-\dfrac{\varepsilon^{2}}{24}-\alpha=0$, $\varepsilon > 0$		$\varphi\in \left(-\dfrac{{\text{π}}}{2},\right. \left.\dfrac{{\text{π}}}{2}\right]+2 n{\text{π}}$	反暗孤子
$\omega^{2}+\dfrac{\varOmega^{2} }{6}-\alpha\neq0, \varOmega\in(0, 2)$		0	$\varphi\in \left(\dfrac{{\text{π}}}{2},\dfrac{3{\text{π}}}{2}\right)+2 n{\text{π}}$	Akhmediev呼吸子
$\omega^{2}+\dfrac{\varOmega^{2} }{6}-\alpha=0$			$0<\|\varOmega\|<\dfrac{\sqrt{3}}{\|\sec\varphi\|}$	W形孤子链
$\omega^{2}+\dfrac{\varOmega^{2} }{6}-\alpha=0$			$\dfrac{\sqrt{3}}{\|\sec\varphi\|}<\|\varOmega\|<\dfrac{2}{\|\sec\varphi\|}$	周期波
$1+2\beta\left(\pm\sqrt{\varDelta}-3\omega\right)+2\gamma\nabla\neq0$			$\varphi\in \mathbb{\rm R}$	Tajiri-Watanabe呼吸子
$1+2\beta\left(\pm\sqrt{\varDelta}-3\omega\right)+2\gamma\nabla=0$			$\varphi\in \mathbb{\rm R}$	多峰孤子
注1: $\omega$, $\varOmega$, $\varepsilon$和$\varphi$分别为背景频率、扰动频率、扰动能量和相对相位. 参数$\alpha=\dfrac{\beta^{2}}{16\gamma^{2}}+\dfrac{1}{12\gamma}+a^{2}$, $\varDelta = {\bigg[ {\dfrac{ {\sqrt { { {({\varepsilon ^2} - 4{\varOmega ^2} + 16{a^2})}^2} + 16{\varepsilon ^2}{\varOmega ^2} } - ({\varepsilon ^2} - 4{\varOmega ^2} + 16{a^2})} }{8} } \bigg]^{1/2} }$, $\nabla=-2\varDelta\pm8\omega\sqrt{\varDelta}-6\omega^{2}+6 a^{2}+\dfrac{1}{4}\varepsilon^{2}-\varOmega^{2}$.

下载: 导出CSV

[1]	Guo B, Pan X 1990 Chin. Phys. Lett. 7 241 Google Scholar
[2]	Lou S Y, Ni G J, Huang G X 1992 Commun. Theor. Phys. 17 67 Google Scholar
[3]	Chabchoub A, Hoffmann N P, Akhmediev N 2011 Phys. Rev. Lett. 106 204502 Google Scholar
[4]	Chabchoub A, Hoffmann N, Onorato M, et al. 2012 Phys. Rev. X 2 011015
[5]	Chabchoub A, Hoffmann N, Onorato M, et al. 2012 Phys. Rev. E 86 056601 Google Scholar
[6]	Chen Z, Segev M, et al 1996 Opt. Lett. 21 1821 Google Scholar
[7]	Chen Z, Segev M, et al 1997 J. Opt. Soc. Am. B 14 1407 Google Scholar
[8]	Guo Q, Luo B, Yi F, Chi S, Xie Y 2004 Phys. Rev. E 69 016602 Google Scholar
[9]	Deng D, Guo Q 2007 Opt. Lett. 32 3206 Google Scholar
[10]	Solli D R, Ropers C, Koonath P, et al. 2007 Nature 450 1054 Google Scholar
[11]	Kibler B, Fatome J, Finot C, et al. 2010 Nat. Phys. 6 790 Google Scholar
[12]	Dudley J M, Genty G, Dias F, et al. 2009 Opt. Express 17 21497 Google Scholar
[13]	Liu X 2011 Phys. Rev. A 84 053828 Google Scholar
[14]	Kibler B, Fatome J, Finot C, et al. 2012 Sci. Rep. 2 463 Google Scholar
[15]	Jia J, Lin J 2012 Opt. Express 20 7469 Google Scholar
[16]	Zhang Y, Belic M, Wu Z, Zheng H, Lu K, Li Y, Zhang Y 2013 Opt. Lett. 38 4585 Google Scholar
[17]	Lin J, Chen W W, Jia J 2014 J. Opt. Soc. Am. A 31 188 Google Scholar
[18]	Liu W, Pang L, Han H, Shen Z, Lei M, Teng H, Wei Z 2016 Photon. Research 4 111 Google Scholar
[19]	Liu W, Pang L, Yan H, Ma G, Lei M, Wei Z 2016 EuroPhys. Lett. 116 64002 Google Scholar
[20]	Liu X, Yao X, Cui Y 2018 Phys. Rev. Lett. 121 023905 Google Scholar
[21]	Liu X, Popa D, Akhmediev N 2019 Phys. Rev. Lett. 123 093901 Google Scholar
[22]	Bailung H, Sharma S K, Nakamura Y 2011 Phys. Rev. Lett. 107 255005 Google Scholar
[23]	Tsai Y Y, Tsai J Y, Lin I 2016 Nat. Phys. 12 573 Google Scholar
[24]	Zhang W, Walls D F 1994 Phys. Rev. Lett. 72 60 Google Scholar
[25]	Zhang W, Walls D F 1994 Phys. Rev. A 49 3799 Google Scholar
[26]	Burger S, Bongs K, Dettmer S, et al. 1999 Phys. Rev. Lett. 83 5198 Google Scholar
[27]	Denschlag J, Simsarian J E, Feder D L, et al. 2000 Science 287 97 Google Scholar
[28]	Huang G X 2001 Chin. Phys. Lett. 18 628 Google Scholar
[29]	Khaykovich L, Schreck F, Ferrari G, et al. 2002 Science 296 1290 Google Scholar
[30]	Strecker K E, Partridge G B, Truscott A G, et al. 2002 Nature 417 150 Google Scholar
[31]	Liang Z X, Zhang Z D, Liu W M 2005 Phys. Rev. Lett. 94 050402 Google Scholar
[32]	Bludov Y V, Konotop V V, Akhmediev N 2009 Phys. Rev. A 80 033610 Google Scholar
[33]	Zhao D, Luo H G, Chai H Y 2008 Phys. Lett. A 372 5644 Google Scholar
[34]	Feng B, Zhao D 2016 J. Differ. Equations 260 2973 Google Scholar
[35]	Zeng J, Malomed B A 2017 Phys. Rev. E 95 052214 Google Scholar
[36]	Yao Y Q, Han W, Li J, Liu W M 2018 J. Phys. B 51 105001 Google Scholar
[37]	Wang D S, Liu J, Wang L 2018 Phys. Lett. A 382 799 Google Scholar
[38]	He Z M, Wen L, Wang Y J, Chen G P, Tan R B, Dai C Q, Zhang X F 2019 Phys. Rev. E 99 062216 Google Scholar
[39]	Shats M, Punzmann H, Xia H 2010 Phys. Rev. Lett. 104 104503 Google Scholar
[40]	Daniel M, Kavitha L, Amuda R 1999 Phys. Rev. B 59 13774 Google Scholar
[41]	Daniel M, Beula J 2009 Chaos, Solitons Fractals 41 1842
[42]	Daniel M, Beula J 2009 Phys. Lett. A 373 2841 Google Scholar
[43]	Zhao F, Li Z D, Li Q Y, et al 2012 Ann. Phys. 327 2085 Google Scholar
[44]	Qi J W, Li Z D, Yang Z Y, et al. 2017 Phys. Lett. A 381 1874 Google Scholar
[45]	Yan Z 2010 Commun. Theor. Phys. 54 947 Google Scholar
[46]	Yan Z 2011 Phys. Lett. A 375 4274 Google Scholar
[47]	Wu Y, Zhao L C, Lei X K 2015 Eur. Phys. J. B 88 297 Google Scholar
[48]	Zheludev N I, Kivshar Y S 2012 Nat. Mater. 11 917 Google Scholar
[49]	Wen S, Wang Y, Su W, Xiang Y, Fu X, Fan D 2006 Phys. Rev. E 73 036617 Google Scholar
[50]	Xiong H, Gan J, Wu Y 2017 Phys. Rev. Lett. 119 153901 Google Scholar
[51]	Konotop V V, Yang J, Zezyulin D A 2016 Rev. Mod. Phys. 88 035002 Google Scholar
[52]	Lou S Y, Huang F 2017 Sci. Rep. 7 869 Google Scholar
[53]	Polo J, Ahufinger V 2013 Phys. Rev. A 88 053628 Google Scholar
[54]	McDonald G D, Kuhn C C N, Hardman K S, et al. 2014 Phys. Rev. Lett. 113 013002 Google Scholar
[55]	Helm J L, Cornish S L, Gardiner S A 2015 Phys. Rev. Lett. 114 134101 Google Scholar
[56]	Zhao L C, Ling L, Yang Z Y, Liu J 2016 Nonlinear Dyn. 83 659 Google Scholar
[57]	Solli D R, Ropers C, Jalali B 2008 Phys. Rev. Lett. 101 233902 Google Scholar
[58]	Frisquet B, Chabchoub A, Fatome J, et al. 2014 Phys. Rev. A 89 023821 Google Scholar
[59]	Gertjerenken B, Billam T P, Blackley C L, et al. 2013 Phys. Rev. Lett. 111 100406 Google Scholar
[60]	Fatome J, Kibler B, Finot C 2013 Opt. Lett. 38 1663 Google Scholar
[61]	Yang G, Wang Y, Qin Z, et al. 2014 Phys. Rev. E 90 062909 Google Scholar
[62]	Zhao L C 2018 Phys. Rev. E 97 062201 Google Scholar
[63]	Hasegawa A, Tappert F 1973 Appl. Phys. Lett. 23 171 Google Scholar
[64]	Zakharov V E, Shabat A B 1973 Sov. Phys. JETP 37 823
[65]	Kivshar Y S, Luther-Davies B 1998 Phys. Rep. 298 81 Google Scholar
[66]	Kivshar Y S 1991 Phys. Rev. A 43 1677 Google Scholar
[67]	Kivshar Y S, Afanasjev V V 1991 Phys. Rev. A 44 R1446 Google Scholar
[68]	Dong G, Liu Z 1996 Opt. Commun. 128 8 Google Scholar
[69]	Li Z, Li L, Tian H, et al. 2000 Phys. Rev. Lett. 84 4096 Google Scholar
[70]	Zhao L C, Li S C, Ling L 2014 Phys. Rev. E 89 023210 Google Scholar
[71]	Liu C, Yang Z Y, Zhao L C, et al. 2015 Phys. Rev. E 91 022904 Google Scholar
[72]	Ren Y, Yang Z Y, Liu C, et al. 2015 Phys. Lett. A 379 2991 Google Scholar
[73]	Liu C, Yang Z Y, Zhao L C, et al. 2016 Phys. Rev. E 94 042221 Google Scholar
[74]	Zhao L C, Li S C, Ling L 2016 Phys. Rev. E 93 032215 Google Scholar
[75]	Peregrine D H 1983 J. Australas. Math. Soc. Ser. B 25 16 Google Scholar
[76]	Akhmediev N N, Korneev V I 1986 Theor. Math. Phys. 1089
[77]	Kuznetsov E A 1977 Akademiia Nauk SSSR Doklady 236 575
[78]	Ma Y C 1979 Stud. Appl. Math. 60 43 Google Scholar
[79]	Tajiri M, Watanabe Y 1998 Phys. Rev. E 57 3510 Google Scholar
[80]	Priya N V, Senthilvelan M, Lakshmanan M 2013 Phys. Rev. E 88 022918 Google Scholar
[81]	刘冲 2016 博士学位论文(西安: 西北大学) Liu C 2016 Ph. D. Dissertation (Xi’an: Northwest University) (in Chinese)
[82]	Chowdury A, Ankiewicz A, Akhmediev N 2015 Proc. R. Soc. A 471 20150130 Google Scholar
[83]	Zakharov V E, Gelash A A 2013 Phys. Rev. Lett. 111 054101 Google Scholar
[84]	Kibler B, Chabchoub A, Gelash A, Akhmediev N, Zakharov V E 2015 Phys. Rev. X 5 041026
[85]	Zhang J H, Wang L, Liu C 2017 Proc. R. Soc. A 473 20160681 Google Scholar
[86]	Liu C, Ren Y, Yang Z Y, Yang W L 2017 Chaos 27 083120 Google Scholar
[87]	Liu C, Yang Z Y, Yang W L 2018 Chaos 28 083110 Google Scholar
[88]	Ren Y, Liu C, Yang Z Y, Yang W L 2018 Phys. Rev. E 98 062223 Google Scholar
[89]	Ren Y, Wang X, Liu C, Yang Z Y, Yang W L 2018 Commun. Nonlinear Sci. Numer. Simul. 63 161 Google Scholar
[90]	Xu G, Gelash A, Chabchoub A, Zakharov V, Kibler B 2019 Phys. Rev. Lett. 122 084101 Google Scholar
[91]	Liu C, Yang Z Y, Yang W L, Akhmediev N 2019 J. Opt. Soc. Am. B 36 1294 Google Scholar
[92]	Guo B, Ling L 2011 Chin. Phys. Lett. 28 110202 Google Scholar
[93]	Guo B, Ling L, Liu Q P 2012 Phys. Rev. E 85 026607 Google Scholar
[94]	Ling L, Zhao L C 2013 Phys. Rev. E 88 043201
[95]	Ling L, Guo B, Zhao L C 2014 Phys. Rev. E 89 041201(R)
[96]	Zhao L C, Liu J 2012 J. Opt. Soc. Am. B 29 3119 Google Scholar
[97]	Zhao L C, Liu J 2013 Phys. Rev. E 87 013201 Google Scholar
[98]	Zhao L C, Xin G G, Yang Z Y 2014 Phys. Rev. E 90 022918 Google Scholar
[99]	Baronio F, Conforti M, Degasperis A, et al. 2013 Phys. Rev. Lett. 111 114101 Google Scholar
[100]	Baronio F, Conforti M, Degasperis A, et al. 2014 Phys. Rev. Lett. 113 034101 Google Scholar
[101]	Zhao L C, Ling L, Yang Z Y, Liu J 2015 Commun. Nonlinear Sci. Numer. Simul. 23 21 Google Scholar
[102]	Chen S, Cai X M, Grelu P, et al. 2016 Opt. Express 24 5886 Google Scholar
[103]	Ling L, Zhao L C, Yang Z Y, et al. 2017 Phys. Rev. E 96 022211 Google Scholar
[104]	Yan Z 2015 Nonlinear Dyn. 79 2515 Google Scholar
[105]	Wen X Y, Yang Y, Yan Z 2015 Phys. Rev. E 92 012917 Google Scholar
[106]	Zhao L C, Guo B, Ling L 2016 J. Math. Phys. 57 043508 Google Scholar
[107]	Zhang G, Yan Z, Wen X Y, Chen Y 2017 Phys. Rev. E 95 042201 Google Scholar
[108]	Li M, Shui J J, Xu T 2018 Appl. Math. Lett. 83 110 Google Scholar
[109]	Zhao L C, Ling L 2014 arXiv: 1410.7536; 2016 J. Opt. Soc. Am. B 33 850
[110]	Baronio F, Chen S, Grelu P, et al. 2015 Phys. Rev. A 91 033804 Google Scholar
[111]	He J S, Xu S, Porsezian K 2012 J. Phys. Soc. Jpn. 81 124007 Google Scholar
[112]	Xu S, He J S, Cheng Y, Porseizan K 2015 Math. Meth. Appl. Sci. 38 1106?Google Scholar
[113]	Xu S, He J S 2012 J. Math. Phys. 53 063507 Google Scholar
[114]	Zhao L C, Yang Z Y, Ling L 2014 J. Phys. Soc. Jpn. 83 104401 Google Scholar
[115]	Zhao L C, Liu C, Yang Z Y 2015 Commun. Nonlinear Sci. Numer. Simul. 20 9 Google Scholar
[116]	Akhmediev N, Soto-Crespo J M, Ankiewicz A 2009 Phys. Rev. A 80 043818 Google Scholar
[117]	He J S, Zhang H R, Wang L H, Porsezian K, Fokas A S 2013 Phys. Rev. E 87 052914 Google Scholar
[118]	Wang L, He J S, Xu H, Wang J, Porsezian K 2017 Phys. Rev. E 95 042217 Google Scholar
[119]	Zakharov V E, Ostrovsky L A 2009 Physica D 238 540 Google Scholar
[120]	Hammani K, Wetzel B, Kibler B, et al. 2011 Opt. Lett. 36 2140 Google Scholar
[121]	Soto-Crespo J M, Ankiewicz A, Devine N, et al. 2012 J. Opt. Soc. Am. B 29 1930 Google Scholar
[122]	Forest M G, McLaughlin D W, Muraki D J, et al. 2000 J. Nonlinear Sci. 10 291 Google Scholar
[123]	Hirota R 1973 J. Math. Phys. 14 805 Google Scholar
[124]	Sasa N, Satsuma J 1991 J. Phys. Soc. Jpn. 60 409 Google Scholar
[125]	Duan L, Zhao L C, Xu W H, et al. 2017 Phys. Rev. E 95 042212 Google Scholar
[126]	Li P, Wang L, Kong L Q, et al. 2018 Appl. Math. Lett. 85 110 Google Scholar
[127]	Erkintalo M, Genty G, Wetzel B, et al. 2011 Phys. Lett. A 375 2029 Google Scholar
[128]	Agrawal G P 2007 Nonlinear Fiber Optics (Massachusetts: Academic Press)
[129]	Kevrekidis P G, Frantzeskakis D, Carretero-Gonzalez R 2007 Emergent Nonlinear Phenomena in BoseEinstein Condensates: Theory and Experiment (New York: Springer Science and Business Media)
[130]	Kawaguchi Y, Ueda M 2012 Phys. Rep. 520 253 Google Scholar
[131]	Gao P, Duan L, Zhao L C, et al. 2019 Chaos 29 083112 Google Scholar
[132]	Duan L, Yang Z Y, Zhao L C, et al. 2016 J. Mod. Opt. 63 1397 Google Scholar
[133]	Liu X S, Zhao L C, Duan L, et al. 2017 Chin. Phys. B 26 120503 Google Scholar
[134]	Liu C, Yang Z Y, Zhao L C, et al. 2015 Ann. Phys. 362 130 Google Scholar
[135]	Wang L, Li S, Qi F H 2016 Nonlinear Dyn. 85 389 Google Scholar
[136]	Wang X, Liu C, Wang L 2017 Chaos 27 093106 Google Scholar
[137]	Blanco-Redondo A, De Sterke C M, Sipe J E, et al. 2016 Nat. Commun. 7 10427 Google Scholar
[138]	Daniel M, Kavitha L 2001 Phys. Rev. B 63 172302 Google Scholar
[139]	Tao Y, He J S 2012 Phys. Rev. E 85 026601 Google Scholar
[140]	Wang L H, Porsezian K, He J S 2013 Phys. Rev. E 87 053202 Google Scholar
[141]	Ankiewicz A, Wang Y, Wabnitz S, et al. 2014 Phys. Rev. E 89 012907 Google Scholar
[142]	Chowdury A, Kedziora D J, Ankiewicz A, et al. 2014 Phys. Rev. E 90 032922 Google Scholar
[143]	Yang Y, Yan Z, Malomed B A 2015 Chaos 25 103112 Google Scholar
[144]	Chowdury A, Kedziora D J, Ankiewicz A, et al. 2015 Phys. Rev. E 91 032928 Google Scholar
[145]	Ankiewicz A, Kedziora D J, Chowdury A, et al. 2016 Phys. Rev. E 93 012206 Google Scholar
[146]	Wang L, Zhang J H, Wang Z Q, et al. 2016 Phys. Rev. E 93 012214 Google Scholar
[147]	Zhao L C, Ling L, Yang Z Y 2018 Phys. Rev. E 97 022218 Google Scholar
[148]	Duan L, Yang Z Y, Gao P, et al. 2019 Phys. Rev. E 99 012216 Google Scholar
[149]	Wen X Y, Yan Z, Boris A, Malomed 216 Chaos 26 123110
[150]	Wen X Y, Yan Z 2018 J. Math. Phys. 59 073511 Google Scholar
[151]	Wen X Y, Yan Z 2017 Commun. Nonlinear. Sci. Numer. Simul. 43 311 Google Scholar
[152]	Ohta Y, Yang J 2012 Phys. Rev. E 86 036604 Google Scholar
[153]	Yang B, Chen Y 2018 Appl. Math. Lett. 82 43 Google Scholar
[154]	Li M, Xu T, Meng D 2016 J. Phys. Soc. Jpn. 85 124001 Google Scholar
[155]	Xu T, Li H, Zhang H, et al. 2017 Appl. Math. Lett. 63 88 Google Scholar
[156]	Dai C Q, Zhu H P 2013 J. Opt. Soc. Am. B 30 3291 Google Scholar
[157]	Chen J, Chen Y, Feng B F, Maruno K 2015 Phys. Lett. A 379 1510 Google Scholar
[158]	Zhang X, Chen Y, Tang X 2018 Comput. Math. Appl. 76 1938 Google Scholar
[159]	Qian C, Rao J G, Liu Y B, He J S 2016 Chin. Phys. Lett. 33 110201 Google Scholar
[160]	Zhang X, Chen Y 2017 Commun. Nonlinear. Sci. Numer. Simul. 52 24 Google Scholar
[161]	Liu Y K, and Li B 2017 Chin. Phys. Lett. 34 010202 Google Scholar
[162]	Charalampidis E G, Wang W, Kevrekidis P G, Frantzeskakis D J, Cuevas-Maraver J 2016 Phys. Rev. A 93 063623 Google Scholar
[163]	Wang W, Kevrekidis P G, Carretero-Gonzalez R, Frantzeskakis D J 2016 Phys. Rev. A 93 023630 Google Scholar
[164]	Rao J, Porsezian K, He J S 2017 Chaos 27 083115??Google Scholar
[165]	Rao J, Porsezian K, He J S, Kanna T 2018 Proc. R. Soc. A 474 20170627
[166]	Zeng L, Zeng J, Kartashov Y V, Malomed B A 2019 Opt. Lett. 44 1206 Google Scholar

[1]	胡智, 李金花, 李萌萌, 马佑桥, 任海东. 非线性光纤中Fermi-Pasta-Ulam-Tsingou现象的稳定性分析. 物理学报, 2024, 73(23): 235201. doi: 10.7498/aps.73.20241380
[2]	郑州, 李金花, 马佑桥, 任海东. 扰动振幅和扰动频率对Fermi-Pasta-Ulam-Tsingou回归现象的影响. 物理学报, 2022, 71(18): 185201. doi: 10.7498/aps.71.20220945
[3]	李再东, 郭奇奇. 铁磁纳米线中磁化强度的磁怪波. 物理学报, 2020, 69(1): 017501. doi: 10.7498/aps.69.20191352
[4]	裴世鑫, 徐辉, 孙婷婷, 李金花. 正三角型三芯光纤中等腰对称平面波的调制不稳定性分析. 物理学报, 2018, 67(5): 054203. doi: 10.7498/aps.67.20171650
[5]	陈海军. 变分法研究二维光晶格中玻色-爱因斯坦凝聚的调制不稳定性. 物理学报, 2015, 64(5): 054702. doi: 10.7498/aps.64.054702
[6]	藤斐, 谢征微. 光晶格中双组分玻色-爱因斯坦凝聚系统的调制不稳定性. 物理学报, 2013, 62(2): 026701. doi: 10.7498/aps.62.026701
[7]	毛杰健, 杨建荣. 大尺度浅水波方程中相互调制的非线性波. 物理学报, 2013, 62(13): 130205. doi: 10.7498/aps.62.130205
[8]	张立升, 邓敏艺, 孔令江, 刘慕仁, 唐国宁. 用元胞自动机模型研究二维激发介质中的非线性波. 物理学报, 2009, 58(7): 4493-4499. doi: 10.7498/aps.58.4493
[9]	黄劲松, 陈海峰, 谢征微. 光晶格中双组分偶极玻色-爱因斯坦凝聚体的调制不稳定性. 物理学报, 2008, 57(6): 3435-3439. doi: 10.7498/aps.57.3435
[10]	丁万山, 席崚, 柳莲花. 基于复Ginzburg-Landau方程的双核光纤中调制不稳定性的仿真研究. 物理学报, 2008, 57(12): 7705-7711. doi: 10.7498/aps.57.7705
[11]	戴小玉, 文双春, 项元江. 色散磁导率对异向介质中的调制不稳定性的影响. 物理学报, 2008, 57(1): 186-193. doi: 10.7498/aps.57.186
[12]	詹杰民, 林东, 李毓湘. 线性与非线性波的Chebyshev广义有限谱模拟. 物理学报, 2007, 56(7): 3649-3654. doi: 10.7498/aps.56.3649
[13]	赵兴东, 谢征微, 张卫平. 玻色凝聚的原子自旋链中的非线性自旋波. 物理学报, 2007, 56(11): 6358-6366. doi: 10.7498/aps.56.6358
[14]	贾维国, 史培明, 杨性愉, 张俊萍, 樊国梁. 高斯变迹布拉格光纤光栅中的调制不稳定性. 物理学报, 2007, 56(9): 5281-5286. doi: 10.7498/aps.56.5281
[15]	刘志芳, 张善元. 圆杆波导中的一个非线性波动方程及准确周期解. 物理学报, 2006, 55(2): 628-633. doi: 10.7498/aps.55.628
[16]	贾维国, 史培明, 杨性愉, 张俊萍, 樊国梁. 保偏光纤中相近频率传输区域的调制不稳定性. 物理学报, 2006, 55(9): 4575-4581. doi: 10.7498/aps.55.4575
[17]	吴昆, 吴健, 徐晗, 曾和平. 超短激光脉冲调制上转换放大. 物理学报, 2005, 54(8): 3749-3756. doi: 10.7498/aps.54.3749
[18]	温　磊, 杨建虎, 胡丽丽. 掺Er3+卤碲酸盐玻璃上转换能级寿命和发光性能研究. 物理学报, 2004, 53(12): 4378-4381. doi: 10.7498/aps.53.4378
[19]	李齐良, 朱海东, 唐向宏, 李承家, 王小军, 林理彬. 有源光放大器链路中交叉相位调制的不稳定性. 物理学报, 2004, 53(12): 4194-4201. doi: 10.7498/aps.53.4194
[20]	张介秋, 梁昌洪, 王耕国, 朱家珍. 阿尔芬高斯波包演化为阿尔芬孤波的条件及阿尔芬波的调制不稳定性判据. 物理学报, 2003, 52(4): 890-895. doi: 10.7498/aps.52.890

计量

文章访问数: 11250
PDF下载量: 324
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

搜索

留言板