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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)
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Google Scholar
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[5] Bludov Yu V, Konotop V V, Akhmediev N 2009 Phys. Rev. A 80 033610
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Google Scholar
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Google Scholar
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Google Scholar
[19] Akhmediev N, Ankiewicz A, Soto Crespo J M 2009 Phys. Rev. E 80 026601
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[49] Pelinovsky D E, Stepanyants Y A, Kivshar Y A 1995 Phys. Rev. E 51 5016
Google Scholar
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Google Scholar
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Google Scholar
[53] Pelinovsky D E, Stepanyants Y A 1993 JETP Lett. 57 24
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[58] Kodama Y 2004 J. Phys. A: Math. Gen. 37 11169
Google Scholar
[59] Biondini G 2007 Phys. Rev. Lett. 99 064103
Google Scholar
[60] Ma W X 2015 Phys. Lett. A 379 1975
Google Scholar
[61] Singh N, Stepanyants Y 2016 Wave Motion 64 92
Google Scholar
[62] Hu W C, Huang W H, L u, Z M, Stepanyants Y 2018 Wave Motion 77 243
Google Scholar
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Google Scholar
[64] Yang J Y, Ma W X 2017 Nonlinear Dyn. 89 1539
Google 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 074102
Google Scholar
[68] Yan Z Y, Zhang X F, Liu W M 2011 Phys. Rev. A 84 023627
Google Scholar
[69] Lou H G, Zhao D, He X 2009 , Phys. Rev. A 79 063802
Google Scholar
[70] Zhang J F, Li Y S, Meng J P, Wu L, Malomed B A 2010 Phys. Rev. A 82 033614
Google Scholar
[71] Dai C Q, Zhang J F 2010 Opt. Lett. 35 2651
Google Scholar
[72] Serkin V N, Hasegawa A, Belyaeva T L 2010 Phys. Rev. A 81 023610
Google Scholar
[73] Kibler B, Fatome J, Finot C, et al. 2010 Nat. Phys. 6 790
Google Scholar
[74] Wu L, Zhang J F, Li L, Tian Q, Porsezian K 2008 Opt. Express 16 6352
Google Scholar
[75] Tian Q, Wu L, Zhang J F, Malomed B A, Mihalache D, Liu W M 2011 Phys. Rev. E 83 016602
Google Scholar
[76] David D, Levi D, Wintemitz P 1987 Stud. Appl. Math. 76 133
Google Scholar
[77] Chan W L, Li K S, Li Y S 1992 J. Math. Phys. 33 3759
Google Scholar
[78] Lü Z S, Chen Y N 2015 Eur. Phys. J. B 88 187
Google Scholar
[79] Ilhan O A, Manafian J, Shahriaric M 2019 Comput. Math. App. 78 2429
Google 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)$t \!=\! 8$ Figure 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)$t = 8$ Figure 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)$t = 8$ Figure 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)$t = 8$ Figure 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)$t = 8$ Figure 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${\upsilon _y} = 0, {\upsilon _y} = - 2, {\upsilon _y} = 2$ )Figure 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)$t = 4$ Figure 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)$t = 4$ Figure 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
Figure 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 5831
Google 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 034801
Google Scholar
[4] Akhmediev N, Dudley J M, Solli D R, Turitsyn S K 2013 J. Opt. 15 060201
Google Scholar
[5] Bludov Yu V, Konotop V V, Akhmediev N 2009 Phys. Rev. A 80 033610
Google Scholar
[6] Moslem W M 2011 Phys. Plasm. 18 032301
Google Scholar
[7] Stenflo L, Marklund M 2010 J. Plasm. Phys. 76 293
Google Scholar
[8] Onorato M, Residori S, Bortolozzo U, Montina A, Arecchi F 2013 Phys. Rep. 528 47
Google Scholar
[9] Müller P, Garrett C, Osborne A 2005 Oceanography 18 66
Google Scholar
[10] 张解放, 戴朝卿 2016 物理学报 65 050501
Google Scholar
Zhang J F, Dai C Q 2016 Acta Phys. Sin 65 050501
Google Scholar
[11] Hohmann R, Kuhl U, Stockmann H J, Kaplan L, Heller E J 2010 Phys. Rev. Lett. 104 093901
Google Scholar
[12] Ganshin A N, Efimov V B, Kolmakov G V, Mezhov Deglin P V, McClintock E 2008 Phys. Rev. Lett. 101 065303
Google Scholar
[13] Yan Z Y 2010 Commun. Theor. Phys. 54 947
Google Scholar
[14] Shats M, Punzmann H, Xia H 2010 Phys. Rev. Lett. 104 104503
Google Scholar
[15] Xia H, Maimbourg T, Punzmann H, Shats M 2012 Phys. Rev. Lett. 109 114502
Google Scholar
[16] Solli R, Ropers C, Koonath P, Jalali B 2007 Nature 450 1054
Google Scholar
[17] Chabchoub A, Hoffmann N P, Akhmediev N 2011 Phys. Rev. Lett. 106 204502
Google Scholar
[18] Peregrine D H 1983 J. Aust. Math. Soc. Ser. B: Appl. Math. 25 16
Google Scholar
[19] Akhmediev N, Ankiewicz A, Soto Crespo J M 2009 Phys. Rev. E 80 026601
Google Scholar
[20] Kedziora D J, Ankiewicz A, Akhmediev N 2012 Phys. Rev. E 86 056602
Google Scholar
[21] Ohta Y, Yang J 2012 Proc. R. Soc. A 468 1716
Google Scholar
[22] Ankiewicz A, Soto Crespo J M, Akhmediev N 2010 Phys. Rev. E 81 046602
Google Scholar
[23] Li L J, Wu Z W, Wang J H, He J S 2013 Annals of Physics 334 198
Google Scholar
[24] Tao Y S, He J S 2012 Phys. Rev. E 85 026601
Google Scholar
[25] Chen S 2013 Phys. Rev. E 88 023202
Google Scholar
[26] Chan H N, Chow K W, Kedziora D J, Grimshaw R H J, Ding E 2014 Phys. Rev. E 89 032914
Google Scholar
[27] Zhang Y S, Guo L J, He J S 2015 Lett. Math. Phys. 105 853
Google Scholar
[28] Qiu D Q, He J, Zhang Y H, Porsezian K 2015 Proc. R. Soc. A 471 20150236
Google Scholar
[29] He J S, Xu S W, Porsezian K 2012 J. Phs. Soc. Japan 81 124007
Google Scholar
[30] Xu S W, He J S, Cheng Y, Porseizan K 2015 Math. Meth. Appli. Sci. 38 1106
Google Scholar
[31] Chen S, Song L Y 2014 Phys. Lett. A 378 1228
Google Scholar
[32] He J S, Wang L, Li L, Porsezian K, Erdélyi R 2014 Phys. Rev. E 89 062917
Google Scholar
[33] Zha Q 2013 Phys. Scr. 87 065401
Google Scholar
[34] Chen S, Soto Crespo J M, Baronio F, Grelu Ph, Mihalache D 2016 Opt. Express 24 15251
Google Scholar
[35] Wang L H, Porsezian K, He J S 2013 Phys. Rev. E 87 053202
Google Scholar
[36] Chen S, Mihalache D 2015 J. Phys. A: Math. Theor. 48 215202
Google Scholar
[37] Baronio F, Conforti M, Degasperis A, Lombardo S, Onorato M, Wabnitz S 2014 Phys. Rev. Lett. 113 034101
Google Scholar
[38] He J S, Zhang H R, Wang L H, Porsezian K, Fokas A S 2013 Phys. Rev. E 87 052914
Google Scholar
[39] Wang L H, He J S, Xu H, Wang J, Porsezian K 2017 Phys. Rev. E 95 042217
Google Scholar
[40] Ohta Y, Yang J 2012 Phys. Rev. E 86 036604
Google Scholar
[41] Ohta Y, Yang J 2013 J. Phys. A: Math. Theor. 46 105202
Google Scholar
[42] Rao J G, Porsezian K, He J S 2017 Chaos 27 083115
Google Scholar
[43] Guo L J, He J S, Wang L H, Cheng Y, Frantzeskakis D J, Kevrekidis P G 2020 Phys. Rev. Res. 2 033376
Google Scholar
[44] Wen L L, Zhang H Q 2016 Nonlinear Dyn. 86 877
Google Scholar
[45] Qiu D Q, Zhang Y S, He J S 2016 Commun. Nonlinear Sci. Numer. Simulat. 30 307
Google Scholar
[46] Jia R R, Guo R 2019 Appl. Math. Lett. 93 117
Google Scholar
[47] Kadomtsev B B, Petviashvili V I 1970 Sov. Phys. Dokl. 15 539
[48] Ablowitz M J, Segur H 1979 J. Fluid Mech. 92 691
Google Scholar
[49] Pelinovsky D E, Stepanyants Y A, Kivshar Y A 1995 Phys. Rev. E 51 5016
Google Scholar
[50] Manakov S V, Zakharov V E, Bordag L A, Matveev V B 1977 Phys. Lett. A 63 205
Google Scholar
[51] Krichever I 1978 Funct. Anal. and Appl. 12 59
[52] Satsuma J, Ablowitz M J 1979 J. Math. Phys. 20 1496
Google Scholar
[53] Pelinovsky D E, Stepanyants Y A 1993 JETP Lett. 57 24
[54] Pelinovsky D E 1994 J. Math. Phys. 35 5820
Google Scholar
[55] Ablowitz M J, Villarroel J 1997 Phys. Rev. Lett. 78 570
Google Scholar
[56] Villarroel J, Ablowitz M J 1999 Comm. Math. Phys. 207 1
Google Scholar
[57] Biondini G, Kodama Y 2003 J. Phys. A: Math. Gen. 36 10519
Google Scholar
[58] Kodama Y 2004 J. Phys. A: Math. Gen. 37 11169
Google Scholar
[59] Biondini G 2007 Phys. Rev. Lett. 99 064103
Google Scholar
[60] Ma W X 2015 Phys. Lett. A 379 1975
Google Scholar
[61] Singh N, Stepanyants Y 2016 Wave Motion 64 92
Google Scholar
[62] Hu W C, Huang W H, L u, Z M, Stepanyants Y 2018 Wave Motion 77 243
Google Scholar
[63] Wen X Y, Yan Z Y 2017 Commun. Nonlinear Sci. Numer. Simulat. 43 311
Google Scholar
[64] Yang J Y, Ma W X 2017 Nonlinear Dyn. 89 1539
Google 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 074102
Google Scholar
[68] Yan Z Y, Zhang X F, Liu W M 2011 Phys. Rev. A 84 023627
Google Scholar
[69] Lou H G, Zhao D, He X 2009 , Phys. Rev. A 79 063802
Google Scholar
[70] Zhang J F, Li Y S, Meng J P, Wu L, Malomed B A 2010 Phys. Rev. A 82 033614
Google Scholar
[71] Dai C Q, Zhang J F 2010 Opt. Lett. 35 2651
Google Scholar
[72] Serkin V N, Hasegawa A, Belyaeva T L 2010 Phys. Rev. A 81 023610
Google Scholar
[73] Kibler B, Fatome J, Finot C, et al. 2010 Nat. Phys. 6 790
Google Scholar
[74] Wu L, Zhang J F, Li L, Tian Q, Porsezian K 2008 Opt. Express 16 6352
Google Scholar
[75] Tian Q, Wu L, Zhang J F, Malomed B A, Mihalache D, Liu W M 2011 Phys. Rev. E 83 016602
Google Scholar
[76] David D, Levi D, Wintemitz P 1987 Stud. Appl. Math. 76 133
Google Scholar
[77] Chan W L, Li K S, Li Y S 1992 J. Math. Phys. 33 3759
Google Scholar
[78] Lü Z S, Chen Y N 2015 Eur. Phys. J. B 88 187
Google Scholar
[79] Ilhan O A, Manafian J, Shahriaric M 2019 Comput. Math. App. 78 2429
Google Scholar
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