# zbMATH — the first resource for mathematics

Characteristics of the breather and rogue waves in a $$(2+1)$$-dimensional nonlinear Schrödinger equation. (English) Zbl 1392.35296
Summary: Under investigation in this paper is a $$(2+1)$$-dimensional nonlinear Schrödinger equation (NLS), which is a generalisation of the NLS equation. By virtue of Wronskian determinants, an effective method is presented to succinctly construct the breather wave and rogue wave solutions of the equation. Furthermore, the main characteristics of the breather and rogue waves are graphically discussed. The results show that rogue waves can come from the extreme behavior of the breather waves. It is hoped that our results could be useful for enriching and explaining some related nonlinear phenomena.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q51 Soliton equations 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
Full Text:
##### References:
 [1] k1 L. Draper, Freak ocean waves, Weather, 21 (1966), 2–4. [2] k3 P. M\"uller and C. Garrett, Rogue waves, Oceanography 18 (2005), 66–75. [3] Ohta, Yasuhiro; Yang, Jianke, General high-order rogue waves and their dynamics in the nonlinear Schr\"odinger equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 468, 2142, 1716-1740, (2012) · Zbl 1364.76033 [4] Kharif, Christian; Pelinovsky, Efim; Slunyaev, Alexey, Rogue waves in the ocean, Advances in Geophysical and Environmental Mechanics and Mathematics, xiv+216 pp., (2009), Springer-Verlag, Berlin · Zbl 1230.86001 [5] Ankiewicz, Adrian; Soto-Crespo, J. M.; Akhmediev, Nail, Rogue waves and rational solutions of the Hirota equation, Phys. Rev. E (3), 81, 4, 046602, 8 pp., (2010) [6] k8 D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, Rogue waves and rational solutions of the nonlinear Schr\"odinger equation, Nature 450, 1054 (2007). [7] Onorato, M.; Residori, S.; Bortolozzo, U.; Montina, A.; Arecchi, F. T., Rogue waves and their generating mechanisms in different physical contexts, Phys. Rep., 528, 2, 47-89, (2013) [8] k12 Z. Y. Yan, Vector financial rogue waves, Phys. Lett. 375 (2011), 4274–4279. · Zbl 1254.91190 [9] k14 J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, Instabilities, breathers and rogue waves in optics, Nature Photonics, 8 (2014), 755–764. [10] k15 J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, Akhmediev breathers and continuous wave supercontinuum generation, Opt. Expr., 17 (2009), 21497–21508. [11] k17 A. N. Ganshin, V. B. Efimov, G. V. Kolmakov, L. P. Mezhov-Deglin, and P. V. E. McClintock, Observation of an inverse energy cascade in developed acoustic turbulence in superfluid helium, Phys. Rev. Lett., 101 (2008), 065303. [12] k19 L. Stenflo and M. Marklund, Rogue waves in the atmosphere, J. Plasma Physics, 76 (2010), 293–295. [13] k20 A. Chabchoub, N. Hoffmann, M. Onorato, A. Slunyaev, A. Sergeeva, E. Pelinovsky, and N. Akhmediev, Observation of a hierarchy of up to fifth-order rogue waves in a water tank, Phys. Rev. E, 86 (2012), 056601. [14] Ablowitz, Mark J., Nonlinear dispersive waves, Cambridge Texts in Applied Mathematics, xiv+348 pp., (2011), Cambridge University Press, New York · Zbl 1232.35002 [15] k23 D. J. Benney and G. J. Roskes, Waves instabilities, Stud. Appl. Math., 48 (1969), 377–385. · Zbl 0216.52904 [16] k24 A. Hasegawa and F. D. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibres, I. Anoma- lous dispersion, Appl. Phys. Lett., 23 (1973), 142–144. [17] Ma, Wen-Xiu; Chen, Min, Direct search for exact solutions to the nonlinear Schr\"odinger equation, Appl. Math. Comput., 215, 8, 2835-2842, (2009) · Zbl 1180.65130 [18] Ma, Wen-Xiu; You, Yuncheng, Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions, Trans. Amer. Math. Soc., 357, 5, 1753-1778, (2005) · Zbl 1062.37077 [19] Ma, Wen-Xiu; Li, Chun-Xia; He, Jingsong, A second Wronskian formulation of the Boussinesq equation, Nonlinear Anal., 70, 12, 4245-4258, (2009) · Zbl 1159.37425 [20] k25 B. L. Guo, L. M. Ling, and Q. P. Liu, Nonlinear Schr\"odinger equation: generalized Darboux transformation and rogue wave solutions, Phys. Rev. E 85 (2012), 026607. [21] k27 U. Bandelow and N. Akhmediev, Persistence of rogue waves in extended nonlinear Schr\"odinger equations: Integrable Sasa-Satsuma case, Phys. Lett. A 376(2012), 1558–1561. · Zbl 1260.35195 [22] Wang, Xin; Li, Yuqi; Chen, Yong, Generalized Darboux transformation and localized waves in coupled Hirota equations, Wave Motion, 51, 7, 1149-1160, (2014) · Zbl 07027081 [23] He, J. S.; Tao, Y. S.; Porsezian, K.; Fokas, A. S., Rogue wave management in an inhomogeneous nonlinear fibre with higher order effects, J. Nonlinear Math. Phys., 20, 3, 407-419, (2013) [24] Zhaolc L. C. Zhao, S. C. Li, and L. M. Ling, Rational W-shaped solitons on a continuous-wave background in the Sasa-Satsuma equation, Phys. Rev. E 89 (2014), 023210. [25] Feng, Lian-Li; Tian, Shou-Fu; Wang, Xiu-Bin; Zhang, Tian-Tian, Rogue waves, homoclinic breather waves and soliton waves for the $$(2+1)$$-dimensional B-type Kadomtsev-Petviashvili equation, Appl. Math. Lett., 65, 90-97, (2017) · Zbl 1355.35034 [26] Wang, Xiu-Bin; Tian, Shou-Fu; Qin, Chun-Yan; Zhang, Tian-Tian, Dynamics of the breathers, rogue waves and solitary waves in the (2+1)-dimensional Ito equation, Appl. Math. Lett., 68, 40-47, (2017) · Zbl 1362.35086 [27] Wang, Mingliang, Erratum: “Solitary wave solutions for variant Boussinesq equations” [Phys. Lett. A \bf199 (1995), no. 3-4, 169–172], Phys. Lett. A, 212, 6, 353 pp., (1996) [28] Akhmediev, N.; Soto-Crespo, J. M.; Ankiewicz, A., Extreme waves that appear from nowhere: on the nature of rogue waves, Phys. Lett. A, 373, 25, 2137-2145, (2009) · Zbl 1229.76012 [29] Ismail, Mourad E. H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications 98, xviii+708 pp., (2009), Cambridge University Press, Cambridge · Zbl 1172.42008 [30] Ismail, Mourad E. H.; Koelink, Erik, Spectral analysis of certain Schr\"odinger operators, SIGMA Symmetry Integrability Geom. Methods Appl., 8, Paper 061, 19 pp., (2012) · Zbl 1270.30003 [31] Ismail, Mourad E. H.; Koelink, Erik, Spectral properties of operators using tridiagonalization, Anal. Appl. (Singap.), 10, 3, 327-343, (2012) · Zbl 1248.33019 [32] k34 Y. Tao and J. He, Multisolitons, breathers, and rogue waves for the Hirota equation generated by the Darboux transformation, Phys. Rev. E, 85 (2012), 026601. [33] k355 C. Bayindir, Rogue waves of the Kundu-Eckhaus equation in a chaotic wave field, Phys. Rev. E, 93 (2016), 032201. [34] k35 Y. Ohta and J. Yang, Rogue waves in the Davey-Stewartson equation, Phys. Rev. E, 86 (2012), 2386–2398. [35] Zhang, Yongshuai; Guo, Lijuan; Xu, Shuwei; Wu, Zhiwei; He, Jingsong, The hierarchy of higher order solutions of the derivative nonlinear Schr\"odinger equation, Commun. Nonlinear Sci. Numer. Simul., 19, 6, 1706-1722, (2014) [36] kph-1 Y. I. Kruglov, A. C. Peacock, and J. D. Harvey, Exact self-similar solutions of the generalized nonlinear Schr\"odinger equation with distributed coefficients, Phys. Rev. Lett. 90 (2003), 113902. [37] Kruglov, V. I.; Peacock, A. C.; Harvey, J. D., Exact solutions of the generalized nonlinear Schr\"odinger equation with distributed coefficients, Phys. Rev. E (3), 71, 5, 056619, 11 pp., (2005) [38] ss-1 E. Suazo and S. K. Suslov, An integral form of the nonlinear Schr\"odinger equation with variable coefficients, (2008), arXiv:08050633v2 [math-ph]. [39] ss-2 E. Suazo and S. K. Suslov, Soliton-like solutions for nonlinear Schr\"odinger equation with variable quadratic Hamiltonians, arXiv:1010.2504v4 [math-ph], [40] Suslov, Sergei K., On integrability of nonautonomous nonlinear Schr\"odinger equations, Proc. Amer. Math. Soc., 140, 9, 3067-3082, (2012) · Zbl 1291.35364 [41] Tian, Shou-Fu, Initial-boundary value problems for the general coupled nonlinear Schr\"odinger equation on the interval via the Fokas method, J. Differential Equations, 262, 1, 506-558, (2017) · Zbl 1432.35194 [42] Tian, Shou-Fu, The mixed coupled nonlinear Schr\"odinger equation on the half-line via the Fokas method, Proc. A., 472, 2195, 20160588, 22 pp., (2016) · Zbl 1371.35278 [43] Kundu, Anjan; Mukherjee, Abhik; Naskar, Tapan, Modelling rogue waves through exact dynamical lump soliton controlled by ocean currents, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470, 2164, 20130576, 20 pp., (2014) · Zbl 1371.86010 [44] Strachan, I. A. B., Wave solutions of a $$(2+1)$$-dimensional generalization of the nonlinear Schr\"odinger equation, Inverse Problems, 8, 5, L21-L27, (1992) · Zbl 0754.35165 [45] Radha, R.; Lakshmanan, M., Singularity structure analysis and bilinear form of a $$(2+1)$$-dimensional non-linear Schr\"odinger (NLS) equation, Inverse Problems, 10, 4, L29-L33, (1994) · Zbl 0809.35127 [46] Dubard, P.; Matveev, V. B., Multi-rogue waves solutions: from the NLS to the KP-I equation, Nonlinearity, 26, 12, R93-R125, (2013) · Zbl 1286.35226 [47] Matveev, V. B.; Smirnov, A. O., Solutions of the Ablowitz-Kaup-Newell-Segur hierarchy equations of the “rogue wave” type: a unified approach, Teoret. Mat. Fiz.. Theoret. and Math. Phys., 186 186, 2, 156-182, (2016) · Zbl 1342.35348 [48] Dubard, P.; Matveev, V. B., Multi-rogue waves solutions: from the NLS to the KP-I equation, Nonlinearity, 26, 12, R93-R125, (2013) · Zbl 1286.35226 [49] Peregrine, D. H., Water waves, nonlinear Schr\"odinger equations and their solutions, J. Austral. Math. Soc. Ser. B, 25, 1, 16-43, (1983) · Zbl 0526.76018 [50] k51 A. Ankiewicz, D. J. Kedziora and N. Akhmediev, Rogue wave triplets, Phys. Lett. A, 375 (2011), 2782–2785. · Zbl 1250.76031 [51] k52 P. Gaillard, Families of quasi-rational solutions of the NLS equation and multi-rogue waves, J. Phys. A, 44 (2011), 435204. · Zbl 1229.35227 [52] k53 E.A. Kuznetsov, Solitons in a parametrically unstable plasma, Dokl. Akad. Nauk SSSR, 236 (1977), 575–577. [53] Ma, Yan Chow, The perturbed plane-wave solutions of the cubic Schr\"odinger equation, Stud. Appl. Math., 60, 1, 43-58, (1979) · Zbl 0412.35028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.