Rogue Waves in the (2+1)-Dimensional Nonlinear Schr?dinger Equation with a Parity-Time-Symmetric Potential
Yun-Kai Liu, Biao Li**
Ningbo Collaborative Innovation Center of Nonlinear Harzard System of Ocean and Atmosphere, and Department of Mathematics, Ningbo University, Ningbo 315211
Abstract ：The (2+1)-dimension nonlocal nonlinear Schrödinger (NLS) equation with the self-induced parity-time symmetric potential is introduced, which provides spatially two-dimensional analogues of the nonlocal NLS equation introduced by Ablowitz et al . [Phys. Rev. Lett. 110 (2013) 064105]. General periodic solutions are derived by the bilinear method. These periodic solutions behave as growing and decaying periodic line waves arising from the constant background and decaying back to the constant background again. By taking long wave limits of the obtained periodic solutions, rogue waves are obtained. It is also shown that these line rogue waves arise from the constant background with a line profile and disappear into the constant background again in the $(x,y)$ plane.
收稿日期: 2016-08-29
出版日期: 2016-12-29
[1] Bludov Y V, Konotop V V and Akhmediev N 2009 Phys. Rev. A 80 033610 [2] Bludov Y V, Konotop V V and Akhmediev N 2010 Eur. Phys. J. Spec. Top. 185 169 [3] Montina A, Bortolozzo U, Residori S and Arecchi F T 2009 Phys. Rev. Lett. 103 173901 [4] Solli D R, Ropers C, Koonath P and Jalali B 2007 Nature 450 1054 [5] Höhmann R, Kuhl U, Stöckmann H J, Kaplan L and Heller E J 2010 Phys. Rev. Lett. 104 093901 [6] Kharif C, Pelinovsky E and Slunyaev A 2009 Rogue Waves in the Ocean (Berlin: Springer) [7] Ganshin A N, Efimov V B, Kolmakov G V, Mezhov-Deglin L P and McClintock P V E 2008 Phys. Rev. Lett. 101 065303 [8] Moslem W M 2011 Phys. Plasmas 18 032301 [9] Bailung H, Sharma S K and Nakamura Y 2011 Phys. Rev. Lett. 107 255005 [10] Yan Z Y 2010 Commun. Theor. Phys. 54 947 [11] Peregrine D H and Aust J 1983 Math. Soc. B 25 16 [12] Dubard P and Matveev V B 2011 Nat. Hazards Earth. Syst. Sci. 11 667 [13] Akhmediev N, Ankiewicz A and Soto-Crespo J M 2009 Phys. Rev. E 80 026601 [14] Dubard P, Gaillard P, Klein C and Matveev V B 2010 Eur. Phys. J. Spec. Top. 185 247 [15] Ankiewicz A, Kedziora D J and Akhmediev N 2011 Phys. Lett. A 375 2782 [16] Guo B, Ling L and Liu Q P 2012 Phys. Rev. E 85 026607 [17] Kedziora D J, Ankiewicz A and Akhmediev N 2011 Phys. Rev. E 84 056611 [18] Ohta Y and Yang J K 2012 Proc. R. Soc. A 468 1716 [19] He J S, Zhang H R, Wang L H, Porsezian K and Fokas A S 2013 Phys. Rev. E 87 052914 [20] Wang L H, Porsezian K and He J S 2013 Phys. Rev. E 87 053202 [21] Wang X, Li Y Q, Huang F and Chen Y 2015 Commun. Nonlinear Sci. Numer. Simulat. 20 434 [22] Wang X, Cao J L and Chen Y 2015 Phys. Scr. 90 105201 [23] He J S, Xu S W and Porsezian K 2012 Phys. Rev. E 86 066603 [24] Rao J G, Wang L H, Zhang Y and He J S 2015 Commun. Theor. Phys. 64 605 [25] Onorato M, Residori S, Bortolozzo U, Montinad A and Arecchi F T 2013 Phys. Reports 528 47 [26] Ohta Y and Yang J K 2012 Phys. Rev. E 86 036604 [27] Ohta Y and Yang J 2013 J. Phys. A 46 105202 [28] Dudley J M, Dias F, Erkintalo M and Genty G 2014 Nat. Photon. 8 755 [29] Dubard P and Matveev V B 2013 Nonlinearity 26 R93 [30] Chen J C, Chen Y and Feng B F 2015 Phys. Lett. A 379 1510 [31] Mu G and Qin Z Y 2014 Nonlinear Anal. Real World Appl. 18 1 [32] Zhang Y, Sun Y B and Xiang W 2015 Appl. Math. Comput. 263 204 [33] Duan L, Yang Z Y, Liu C and Yang W L 2016 Chin. Phys. Lett. 33 010501 [34] Guo B L and Ling L M 2011 Chin. Phys. Lett. 28 110201 [35] Li Y Q, Chen J C, Chen Y and Lou S Y 2014 Chin. Phys. Lett. 31 010201 [36] Ablowitz M J and Musslimani Z H 2013 Phys. Rev. Lett. 110 064105 [37] Bender C M and Boettcher S 1998 Phys. Rev. Lett. 80 5243 [38] Bender C M, Brody D C and Jones H F 2002 Phys. Rev. Lett. 89 270401 [39] Christodoulides D N and Miri M A 2014 Proc. SPIE 9162 1 [40] El-Ganainy R, Makris K, Christodoulides D N and Musslimani Z H 2007 Opt. Lett. 32 2632 [41] Lévai G and Znojil M 2000 J. Phys. A 33 7165 [42] Berini P and De Leon I 2011 Nat. Photon. 6 16 [43] Makris K, El-Ganainy R, Christodoulides D N and Musslimani Z H 2008 Phys. Rev. Lett. 100 103904 [44] Regensburger A, Bersch C, Miri M A, Onishchukov G, Christodoulides D N and Peschel U 2012 Nature 488 167 [45] Ruter C E, Makris K G, El-Ganainy R, Christodoulides D N, Segev M and Kip D 2010 Nat. Phys. 6 192 [46] Yan Z Y 2015 Appl. Math. Lett. 47 61 [47] Yan Z Y 2016 Appl. Math. Lett. 62 101 [48] Wen X Y, Yan Z Y and Yang Y Q 2016 Chaos 26 063123 [49] Horikis T P and Ablowitz M J 2016 arXiv:1608.00927 [50] Hirota R 2004 The direct method in soliton theory (Cambridge University Press, Cambridge) [51] Tajiri M and Watanabe Y 1998 Phys. Rev. E 57 3510 [52] Tajiri M and Watanabe Y 1999 Phys. Rev. E 60 2297 [53] Xu Z X and Chow K W 2016 Appl. Math. Lett. 56 72
[1]
.
[2]
.
[3]
.
[4]
.
[5]
.
[6]
.
[7]
.
[8]
.
[9]
.
[10]
.
[11]
.
[12]
.
[13]
.
[14]
.
[15]
.
2017, Vol. 34 
