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首先给出非自治Kadomtsev-Petviashvili方程转换为Kadomtsev- Petviashvili方程的一个自相似变换, 然后基于Kadomtsev-Petviashvili方程的Lump解构造了非自治Kadomtsev-Petviashvili方程的有理函数表示的二维单、双、三怪波解, 最后通过合适选取变参数, 用图示说明了它们的演化特征, 并利用快速傅里叶变换算法数值模拟测试了二维单怪波的动力学稳定性. 本文方法对寻找(2 + 1)维非线性波动模型的怪波激发提供了启迪.
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关键词:
- 二维怪波 /
- Kadomtsev-Petviashvili方程 /
- 非自治非线性波动模型 /
- 自相似变换
Rogue wave is a kind of natural phenomenon that is fascinating, rare, and extreme. It has become a frontier of academic research. The rogue wave is considered as a spatiotemporal local rational function solution of nonlinear wave model. There are still very few (2 + 1)-dimensional nonlinear wave models which have rogue wave solutions, in comparison with soliton and Lump waves that are found in almost all (2 + 1)-dimensional nonlinear wave models and can be solved by different methods, such as inverse scattering method, Hirota bilinear method, Darboux transform method, Riemann-Hilbert method, and homoclinic test method. The structure and evolution characteristics of the obtained (2 + 1)-dimensional rogue waves are quite different from the prototypes of the (1 + 1)-dimensional nonlinear Schrödinger equation. Therefore, it is of great value to study two-dimensional rogue waves. In this paper, the non-autonomous Kadomtsev-Petviashvili equation is first converted into the Kadomtsev-Petviashvili equation with the aid of a similar transformation, then two-dimensional rogue wave solutions represented by the rational functions of the non-autonomous Kadomtsev-Petviashvili equation are constructed based on the Lump solution of the first kind of Kadomtsev-Petviashvili equation, and their evolutionary characteristics are illustrated by images through appropriately selecting the variable parameters and the dynamic stability of two-dimensional single rogue waves is numerically simulated by the fast Fourier transform algorithm. The obtained two-dimensional rogue waves, which are localized in both space and time, can be viewed as a two-dimensional analogue to the Peregrine soliton and thus are a natural candidate for describing the rogue wave phenomena. The method presented here provides enlightenment for searching for rogue wave excitation of (2 + 1)-dimensional nonlinear wave models. We show that two-dimensional rogue waves are localized in both space and time which arise from the zero background and then disappear into the zero background again. These rogue-wave solutions to the non-autonomous Kadomtsev-Petviashvili equation generalize the rogue waves of the nonlinear Schrödinger equation into two spatial dimensions, and they could play a role in physically understanding the rogue water waves in the ocean. -
Keywords:
- two-dimensional rogue wave /
- Kadomtsev-Petviashvili equation /
- nonautonomous nonlinear wave model /
- self-similar transformation
[1] Pelinovsky E, Kharf C 2008 Extreme Ocean Waves (Berlin: Springer)
[2] Onorato M, Osborne A R, Serio M, Bertone S 2001 Phys. Rev. Lett. 86 5831Google Scholar
[3] Ginzburg N S, Rozental R M, Sergeev A S, Fedotov A E, Zotova I V, Tarakanov V P 2017 Phys. Rev. Lett. 119 034801Google Scholar
[4] Akhmediev N, Dudley J M, Solli D R, Turitsyn S K 2013 J. Opt. 15 060201Google Scholar
[5] Bludov Yu V, Konotop V V, Akhmediev N 2009 Phys. Rev. A 80 033610Google Scholar
[6] Moslem W M 2011 Phys. Plasm. 18 032301Google Scholar
[7] Stenflo L, Marklund M 2010 J. Plasm. Phys. 76 293Google Scholar
[8] Onorato M, Residori S, Bortolozzo U, Montina A, Arecchi F 2013 Phys. Rep. 528 47Google Scholar
[9] Müller P, Garrett C, Osborne A 2005 Oceanography 18 66Google Scholar
[10] 张解放, 戴朝卿 2016 物理学报 65 050501Google Scholar
Zhang J F, Dai C Q 2016 Acta Phys. Sin 65 050501Google Scholar
[11] Hohmann R, Kuhl U, Stockmann H J, Kaplan L, Heller E J 2010 Phys. Rev. Lett. 104 093901Google Scholar
[12] Ganshin A N, Efimov V B, Kolmakov G V, Mezhov Deglin P V, McClintock E 2008 Phys. Rev. Lett. 101 065303Google Scholar
[13] Yan Z Y 2010 Commun. Theor. Phys. 54 947Google Scholar
[14] Shats M, Punzmann H, Xia H 2010 Phys. Rev. Lett. 104 104503Google Scholar
[15] Xia H, Maimbourg T, Punzmann H, Shats M 2012 Phys. Rev. Lett. 109 114502Google Scholar
[16] Solli R, Ropers C, Koonath P, Jalali B 2007 Nature 450 1054Google Scholar
[17] Chabchoub A, Hoffmann N P, Akhmediev N 2011 Phys. Rev. Lett. 106 204502Google Scholar
[18] Peregrine D H 1983 J. Aust. Math. Soc. Ser. B: Appl. Math. 25 16Google Scholar
[19] Akhmediev N, Ankiewicz A, Soto Crespo J M 2009 Phys. Rev. E 80 026601Google Scholar
[20] Kedziora D J, Ankiewicz A, Akhmediev N 2012 Phys. Rev. E 86 056602Google Scholar
[21] Ohta Y, Yang J 2012 Proc. R. Soc. A 468 1716Google Scholar
[22] Ankiewicz A, Soto Crespo J M, Akhmediev N 2010 Phys. Rev. E 81 046602Google Scholar
[23] Li L J, Wu Z W, Wang J H, He J S 2013 Annals of Physics 334 198Google Scholar
[24] Tao Y S, He J S 2012 Phys. Rev. E 85 026601Google Scholar
[25] Chen S 2013 Phys. Rev. E 88 023202Google Scholar
[26] Chan H N, Chow K W, Kedziora D J, Grimshaw R H J, Ding E 2014 Phys. Rev. E 89 032914Google Scholar
[27] Zhang Y S, Guo L J, He J S 2015 Lett. Math. Phys. 105 853Google Scholar
[28] Qiu D Q, He J, Zhang Y H, Porsezian K 2015 Proc. R. Soc. A 471 20150236Google Scholar
[29] He J S, Xu S W, Porsezian K 2012 J. Phs. Soc. Japan 81 124007Google Scholar
[30] Xu S W, He J S, Cheng Y, Porseizan K 2015 Math. Meth. Appli. Sci. 38 1106Google Scholar
[31] Chen S, Song L Y 2014 Phys. Lett. A 378 1228Google Scholar
[32] He J S, Wang L, Li L, Porsezian K, Erdélyi R 2014 Phys. Rev. E 89 062917Google Scholar
[33] Zha Q 2013 Phys. Scr. 87 065401Google Scholar
[34] Chen S, Soto Crespo J M, Baronio F, Grelu Ph, Mihalache D 2016 Opt. Express 24 15251Google Scholar
[35] Wang L H, Porsezian K, He J S 2013 Phys. Rev. E 87 053202Google Scholar
[36] Chen S, Mihalache D 2015 J. Phys. A: Math. Theor. 48 215202Google Scholar
[37] Baronio F, Conforti M, Degasperis A, Lombardo S, Onorato M, Wabnitz S 2014 Phys. Rev. Lett. 113 034101Google Scholar
[38] He J S, Zhang H R, Wang L H, Porsezian K, Fokas A S 2013 Phys. Rev. E 87 052914Google Scholar
[39] Wang L H, He J S, Xu H, Wang J, Porsezian K 2017 Phys. Rev. E 95 042217Google Scholar
[40] Ohta Y, Yang J 2012 Phys. Rev. E 86 036604Google Scholar
[41] Ohta Y, Yang J 2013 J. Phys. A: Math. Theor. 46 105202Google Scholar
[42] Rao J G, Porsezian K, He J S 2017 Chaos 27 083115Google Scholar
[43] Guo L J, He J S, Wang L H, Cheng Y, Frantzeskakis D J, Kevrekidis P G 2020 Phys. Rev. Res. 2 033376Google Scholar
[44] Wen L L, Zhang H Q 2016 Nonlinear Dyn. 86 877Google Scholar
[45] Qiu D Q, Zhang Y S, He J S 2016 Commun. Nonlinear Sci. Numer. Simulat. 30 307Google Scholar
[46] Jia R R, Guo R 2019 Appl. Math. Lett. 93 117Google Scholar
[47] Kadomtsev B B, Petviashvili V I 1970 Sov. Phys. Dokl. 15 539
[48] Ablowitz M J, Segur H 1979 J. Fluid Mech. 92 691Google Scholar
[49] Pelinovsky D E, Stepanyants Y A, Kivshar Y A 1995 Phys. Rev. E 51 5016Google Scholar
[50] Manakov S V, Zakharov V E, Bordag L A, Matveev V B 1977 Phys. Lett. A 63 205Google Scholar
[51] Krichever I 1978 Funct. Anal. and Appl. 12 59
[52] Satsuma J, Ablowitz M J 1979 J. Math. Phys. 20 1496Google Scholar
[53] Pelinovsky D E, Stepanyants Y A 1993 JETP Lett. 57 24
[54] Pelinovsky D E 1994 J. Math. Phys. 35 5820Google Scholar
[55] Ablowitz M J, Villarroel J 1997 Phys. Rev. Lett. 78 570Google Scholar
[56] Villarroel J, Ablowitz M J 1999 Comm. Math. Phys. 207 1Google Scholar
[57] Biondini G, Kodama Y 2003 J. Phys. A: Math. Gen. 36 10519Google Scholar
[58] Kodama Y 2004 J. Phys. A: Math. Gen. 37 11169Google Scholar
[59] Biondini G 2007 Phys. Rev. Lett. 99 064103Google Scholar
[60] Ma W X 2015 Phys. Lett. A 379 1975Google Scholar
[61] Singh N, Stepanyants Y 2016 Wave Motion 64 92Google Scholar
[62] Hu W C, Huang W H, L u, Z M, Stepanyants Y 2018 Wave Motion 77 243Google Scholar
[63] Wen X Y, Yan Z Y 2017 Commun. Nonlinear Sci. Numer. Simulat. 43 311Google Scholar
[64] Yang J Y, Ma W X 2017 Nonlinear Dyn. 89 1539Google Scholar
[65] Jia M, Lou S 2018 arXiv: 1803.01730 v1[nlin.SI]
[66] Serkin V N, Hasegawa A 2000 Phys. Rev. Lett. 85 4502
[67] Serkin V N, Hasegawa A, Belyaeva T L 2007 Phys. Rev. Lett. 98 074102Google Scholar
[68] Yan Z Y, Zhang X F, Liu W M 2011 Phys. Rev. A 84 023627Google Scholar
[69] Lou H G, Zhao D, He X 2009 , Phys. Rev. A 79 063802Google Scholar
[70] Zhang J F, Li Y S, Meng J P, Wu L, Malomed B A 2010 Phys. Rev. A 82 033614Google Scholar
[71] Dai C Q, Zhang J F 2010 Opt. Lett. 35 2651Google Scholar
[72] Serkin V N, Hasegawa A, Belyaeva T L 2010 Phys. Rev. A 81 023610Google Scholar
[73] Kibler B, Fatome J, Finot C, et al. 2010 Nat. Phys. 6 790Google Scholar
[74] Wu L, Zhang J F, Li L, Tian Q, Porsezian K 2008 Opt. Express 16 6352Google Scholar
[75] Tian Q, Wu L, Zhang J F, Malomed B A, Mihalache D, Liu W M 2011 Phys. Rev. E 83 016602Google Scholar
[76] David D, Levi D, Wintemitz P 1987 Stud. Appl. Math. 76 133Google Scholar
[77] Chan W L, Li K S, Li Y S 1992 J. Math. Phys. 33 3759Google Scholar
[78] Lü Z S, Chen Y N 2015 Eur. Phys. J. B 88 187Google Scholar
[79] Ilhan O A, Manafian J, Shahriaric M 2019 Comput. Math. App. 78 2429Google Scholar
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图 1 由(12)式所确定的非自治KP方程的二维单怪波演化 (a)
$t \!=\! - 6$ ; (b)$t \!=\! - 3$ ; (c)$t \!=\! 0$ ; (d)$t \!=\! 0.5$ ; (e)$t \!=\! 4$ ; (f)$t \!=\! 8$ Fig. 1. Evolution of two-dimensional single rogue wave propagation given in Eq. (12) for non-autonomous KP equation: (a)
$t = - 6$ ; (b)$t = - 3$ ; (c)$t = 0$ ; (d)$t = 0.5$ ; (e)$t = 4$ ; (f)$t = 8$ .图 2 由(13)式所确定的非自治KP方程的二维单怪波演化 (a)
$t = - 6$ ; (b)$t = - 3$ ; (c)$t = 0$ ; (d)$t = 0.5$ ; (e)$t = 4$ ; (f)$t = 8$ Fig. 2. Evolution of two-dimensional single rogue wave propagation given in Eq. (13) for non-autonomous KP equation: (a)
$t = - 6$ ; (b)$t = - 3$ ; (c)$t = 0$ ; (d)$t = 0.5$ ; (e)$t = 4$ ; (f)$t = 8$ .图 3 由(14)式所确定的非自治KP方程的二维单怪波演化 (a)
$t = - 6$ ; (b)$t = - 3$ ; (c)$t = 0$ ; (d)$t = 0.5$ ; (e)$t = 4$ ; (f)$t = 8$ Fig. 3. Evolution of two-dimensional single rogue wave propagation given in Eq. (14) for non-autonomous KP equation: (a)
$t = - 6$ ; (b)$t = - 3$ ; (c)$t = 0$ ; (d)$t = 0.5$ ; (e)$t = 4$ ; (f)$t = 8$ .图 4 由(11)式所确定的非自治KP方程的二维双怪波演化(选取
$k = 1/2, l = 1/2, n = 0, m = 1, \lambda = \varepsilon = 1, \nu = 1, \chi = 0, {\upsilon _x} = 2, {\upsilon _y} = 1, a = b = 0$ ) (a)$t = - 6$ ; (b)$t = - 3$ ; (c)$t = 0$ ; (d)$t = 0.5$ ; (e)$t = 4$ ; (f)$t = 8$ Fig. 4. Time evolution of two-dimensional double rogue waves propagation given in Eq. (11) for non-autonomous KP equation when
$k = 1/2, l = 1/2, n = 0, m = 1, $ $\lambda = \varepsilon = 1, \nu = 1, \chi = 0, {\upsilon _x} = 2, {\upsilon _y} = 1, a = b = 0$ : (a)$t = - 6$ ; (b)$t = - 3$ ; (c)$t = 0$ ; (d)$t = 0.5$ ; (e)$t = 4$ ; (f)$t = 8$ .图 5 由(11)式所确定的非自治KP方程二维三怪波演化(选择
$k = l = 1/2, n = 0, m = 1, $ $\lambda = \varepsilon = 1, \nu = 1, \chi = 0, {\upsilon _x} = 2, {\upsilon _y} = 2, a = 5000, b = 5000$ ) (a)$t = - 6$ ; (b)$t = - 3$ ; (c)$t = 0$ ; (d)$t = 0.5$ ; (e)$t = 4$ ; (f)$t = 8$ Fig. 5. Time evolution of two-dimensional triple rogue waves propagation given in Eq. (11) for non-autonomous KP equation when
$k = l = 1/2, n = 0, m = 1, \lambda = \varepsilon = 1, $ $\nu = 1, \chi = 0, {\upsilon _x} = 2, {\upsilon _y} = 1, a = 5000, b = 5000$ : (a)$t = - 6$ ; (b)$t = - 3$ ; (c)$t = 0$ ; (d)$t = 0.5$ ; (e)$t = 4$ ; (f)$t = 8$ .图 6 由(11)式所确定的二维双、三怪波(选取
$k = 1/2, l = 1/2, n = 0, m = 1, $ $\lambda = 1, \varepsilon = 1, \nu = 1, \chi = 0, {\upsilon _x} = 2$ ,$\mu =0, \mu =0.8165, \mu =-0.8165$ 分别对应选取${\upsilon _y} = 0, {\upsilon _y} = - 2, {\upsilon _y} = 2$ )Fig. 6. Profiles of two-dimensional double and triple rogue waves given in Eq. (11) for non-autonomous KP equation when
$k = 1/2, l = 1/2, n = 0, m = 1, \lambda = 1, \varepsilon = 1, \nu = 1, $ $\chi = 0, {\upsilon _x} = 2$ ,$\mu =0, \mu =0.8165, \mu =-0.8165$ correspond to${\upsilon _y}{{ = 0}}, {\upsilon _y} = - 2, {\upsilon _y} = 2$ , respectively.图 7 加了高斯白噪声扰动后由(15)式所确定的二维单怪波演化 (a)
$t = - 5$ ; (b)$t = - {\rm{3}}$ ; (c)$t = - 1$ ; (d)$t = 0$ ; (e)$t = {2}.5$ ; (f)$t = 4$ Fig. 7. Evolution of two-dimensional single rogue wave determined by Eq. (15) after Gaussian white noise disturbance: (a)
$t = - 5$ ; (b)$t = - {\rm{3}}$ ; (c)$t = - 1$ ; (d)$t = 0$ ; (e)$t = {2}.5$ ; (f)$t = 4$ .图 8 加了高斯白噪声扰动后由(16)式所确定的二维单怪波演化 (a)
$t = - 5$ ; (b)$t = - {\rm{3}}$ , (c)$t = - 1$ ; (d)$t = 0$ ; (e)$t = {2}.5$ ; (f)$t = 4$ Fig. 8. Evolution of two-dimensional single rogue wave determined by Eq. (16) after Gaussian white noise disturbance: (a)
$t = - 5$ ; (b)$t = - {\rm{3}}$ ; (c)$t = - 1$ ; (d)$t = 0$ ; (e)$t = {2}.5$ ; (f)$t = 4$ .图 9 在时间区间[–5, 5] x-y平面上非自治KP方程的二维单怪波最大波动值和最小波动值的解析结果和数值计算模拟的对照图 (a)对应二维单怪波((15)式); (b)对应二维单怪波((16)式); (c)在(a)中加了高斯白噪声扰动; (d)在(b)中加高斯白噪声扰动
Fig. 9. Simulation diagram of the analytic and numerical results of the maximum and minimum fluctuations of two-dimensional single rogue waves for the non- autonomous KP equation in the x-y plane of the time interval [–5, 5]: (a) Corresponds to a two-dimensional single rogue wave (Eq. (15)); (b) Corresponds to a two- dimensional single rogue wave (Eq. (16)); (c) Gaussian white noise is added in panel (a); (d) Gaussian white noise is added in panel (b).
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[1] Pelinovsky E, Kharf C 2008 Extreme Ocean Waves (Berlin: Springer)
[2] Onorato M, Osborne A R, Serio M, Bertone S 2001 Phys. Rev. Lett. 86 5831Google Scholar
[3] Ginzburg N S, Rozental R M, Sergeev A S, Fedotov A E, Zotova I V, Tarakanov V P 2017 Phys. Rev. Lett. 119 034801Google Scholar
[4] Akhmediev N, Dudley J M, Solli D R, Turitsyn S K 2013 J. Opt. 15 060201Google Scholar
[5] Bludov Yu V, Konotop V V, Akhmediev N 2009 Phys. Rev. A 80 033610Google Scholar
[6] Moslem W M 2011 Phys. Plasm. 18 032301Google Scholar
[7] Stenflo L, Marklund M 2010 J. Plasm. Phys. 76 293Google Scholar
[8] Onorato M, Residori S, Bortolozzo U, Montina A, Arecchi F 2013 Phys. Rep. 528 47Google Scholar
[9] Müller P, Garrett C, Osborne A 2005 Oceanography 18 66Google Scholar
[10] 张解放, 戴朝卿 2016 物理学报 65 050501Google Scholar
Zhang J F, Dai C Q 2016 Acta Phys. Sin 65 050501Google Scholar
[11] Hohmann R, Kuhl U, Stockmann H J, Kaplan L, Heller E J 2010 Phys. Rev. Lett. 104 093901Google Scholar
[12] Ganshin A N, Efimov V B, Kolmakov G V, Mezhov Deglin P V, McClintock E 2008 Phys. Rev. Lett. 101 065303Google Scholar
[13] Yan Z Y 2010 Commun. Theor. Phys. 54 947Google Scholar
[14] Shats M, Punzmann H, Xia H 2010 Phys. Rev. Lett. 104 104503Google Scholar
[15] Xia H, Maimbourg T, Punzmann H, Shats M 2012 Phys. Rev. Lett. 109 114502Google Scholar
[16] Solli R, Ropers C, Koonath P, Jalali B 2007 Nature 450 1054Google Scholar
[17] Chabchoub A, Hoffmann N P, Akhmediev N 2011 Phys. Rev. Lett. 106 204502Google Scholar
[18] Peregrine D H 1983 J. Aust. Math. Soc. Ser. B: Appl. Math. 25 16Google Scholar
[19] Akhmediev N, Ankiewicz A, Soto Crespo J M 2009 Phys. Rev. E 80 026601Google Scholar
[20] Kedziora D J, Ankiewicz A, Akhmediev N 2012 Phys. Rev. E 86 056602Google Scholar
[21] Ohta Y, Yang J 2012 Proc. R. Soc. A 468 1716Google Scholar
[22] Ankiewicz A, Soto Crespo J M, Akhmediev N 2010 Phys. Rev. E 81 046602Google Scholar
[23] Li L J, Wu Z W, Wang J H, He J S 2013 Annals of Physics 334 198Google Scholar
[24] Tao Y S, He J S 2012 Phys. Rev. E 85 026601Google Scholar
[25] Chen S 2013 Phys. Rev. E 88 023202Google Scholar
[26] Chan H N, Chow K W, Kedziora D J, Grimshaw R H J, Ding E 2014 Phys. Rev. E 89 032914Google Scholar
[27] Zhang Y S, Guo L J, He J S 2015 Lett. Math. Phys. 105 853Google Scholar
[28] Qiu D Q, He J, Zhang Y H, Porsezian K 2015 Proc. R. Soc. A 471 20150236Google Scholar
[29] He J S, Xu S W, Porsezian K 2012 J. Phs. Soc. Japan 81 124007Google Scholar
[30] Xu S W, He J S, Cheng Y, Porseizan K 2015 Math. Meth. Appli. Sci. 38 1106Google Scholar
[31] Chen S, Song L Y 2014 Phys. Lett. A 378 1228Google Scholar
[32] He J S, Wang L, Li L, Porsezian K, Erdélyi R 2014 Phys. Rev. E 89 062917Google Scholar
[33] Zha Q 2013 Phys. Scr. 87 065401Google Scholar
[34] Chen S, Soto Crespo J M, Baronio F, Grelu Ph, Mihalache D 2016 Opt. Express 24 15251Google Scholar
[35] Wang L H, Porsezian K, He J S 2013 Phys. Rev. E 87 053202Google Scholar
[36] Chen S, Mihalache D 2015 J. Phys. A: Math. Theor. 48 215202Google Scholar
[37] Baronio F, Conforti M, Degasperis A, Lombardo S, Onorato M, Wabnitz S 2014 Phys. Rev. Lett. 113 034101Google Scholar
[38] He J S, Zhang H R, Wang L H, Porsezian K, Fokas A S 2013 Phys. Rev. E 87 052914Google Scholar
[39] Wang L H, He J S, Xu H, Wang J, Porsezian K 2017 Phys. Rev. E 95 042217Google Scholar
[40] Ohta Y, Yang J 2012 Phys. Rev. E 86 036604Google Scholar
[41] Ohta Y, Yang J 2013 J. Phys. A: Math. Theor. 46 105202Google Scholar
[42] Rao J G, Porsezian K, He J S 2017 Chaos 27 083115Google Scholar
[43] Guo L J, He J S, Wang L H, Cheng Y, Frantzeskakis D J, Kevrekidis P G 2020 Phys. Rev. Res. 2 033376Google Scholar
[44] Wen L L, Zhang H Q 2016 Nonlinear Dyn. 86 877Google Scholar
[45] Qiu D Q, Zhang Y S, He J S 2016 Commun. Nonlinear Sci. Numer. Simulat. 30 307Google Scholar
[46] Jia R R, Guo R 2019 Appl. Math. Lett. 93 117Google Scholar
[47] Kadomtsev B B, Petviashvili V I 1970 Sov. Phys. Dokl. 15 539
[48] Ablowitz M J, Segur H 1979 J. Fluid Mech. 92 691Google Scholar
[49] Pelinovsky D E, Stepanyants Y A, Kivshar Y A 1995 Phys. Rev. E 51 5016Google Scholar
[50] Manakov S V, Zakharov V E, Bordag L A, Matveev V B 1977 Phys. Lett. A 63 205Google Scholar
[51] Krichever I 1978 Funct. Anal. and Appl. 12 59
[52] Satsuma J, Ablowitz M J 1979 J. Math. Phys. 20 1496Google Scholar
[53] Pelinovsky D E, Stepanyants Y A 1993 JETP Lett. 57 24
[54] Pelinovsky D E 1994 J. Math. Phys. 35 5820Google Scholar
[55] Ablowitz M J, Villarroel J 1997 Phys. Rev. Lett. 78 570Google Scholar
[56] Villarroel J, Ablowitz M J 1999 Comm. Math. Phys. 207 1Google Scholar
[57] Biondini G, Kodama Y 2003 J. Phys. A: Math. Gen. 36 10519Google Scholar
[58] Kodama Y 2004 J. Phys. A: Math. Gen. 37 11169Google Scholar
[59] Biondini G 2007 Phys. Rev. Lett. 99 064103Google Scholar
[60] Ma W X 2015 Phys. Lett. A 379 1975Google Scholar
[61] Singh N, Stepanyants Y 2016 Wave Motion 64 92Google Scholar
[62] Hu W C, Huang W H, L u, Z M, Stepanyants Y 2018 Wave Motion 77 243Google Scholar
[63] Wen X Y, Yan Z Y 2017 Commun. Nonlinear Sci. Numer. Simulat. 43 311Google Scholar
[64] Yang J Y, Ma W X 2017 Nonlinear Dyn. 89 1539Google Scholar
[65] Jia M, Lou S 2018 arXiv: 1803.01730 v1[nlin.SI]
[66] Serkin V N, Hasegawa A 2000 Phys. Rev. Lett. 85 4502
[67] Serkin V N, Hasegawa A, Belyaeva T L 2007 Phys. Rev. Lett. 98 074102Google Scholar
[68] Yan Z Y, Zhang X F, Liu W M 2011 Phys. Rev. A 84 023627Google Scholar
[69] Lou H G, Zhao D, He X 2009 , Phys. Rev. A 79 063802Google Scholar
[70] Zhang J F, Li Y S, Meng J P, Wu L, Malomed B A 2010 Phys. Rev. A 82 033614Google Scholar
[71] Dai C Q, Zhang J F 2010 Opt. Lett. 35 2651Google Scholar
[72] Serkin V N, Hasegawa A, Belyaeva T L 2010 Phys. Rev. A 81 023610Google Scholar
[73] Kibler B, Fatome J, Finot C, et al. 2010 Nat. Phys. 6 790Google Scholar
[74] Wu L, Zhang J F, Li L, Tian Q, Porsezian K 2008 Opt. Express 16 6352Google Scholar
[75] Tian Q, Wu L, Zhang J F, Malomed B A, Mihalache D, Liu W M 2011 Phys. Rev. E 83 016602Google Scholar
[76] David D, Levi D, Wintemitz P 1987 Stud. Appl. Math. 76 133Google Scholar
[77] Chan W L, Li K S, Li Y S 1992 J. Math. Phys. 33 3759Google Scholar
[78] Lü Z S, Chen Y N 2015 Eur. Phys. J. B 88 187Google Scholar
[79] Ilhan O A, Manafian J, Shahriaric M 2019 Comput. Math. App. 78 2429Google Scholar
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