Higher-order rational solutions for a new integrable nonlocal fifth-order nonlinear Schrödinger equation. (English) Zbl 1524.35613
Summary: We study the soliton and rational solutions of a new integrable nonlocal fifth-order nonlinear Schrödinger (NFONLS) equation with three free parameters. In particular, the nonlocal classical NLS, nonlocal Hirota and nonlocal Lakshmanan-Porsezian-Daniel (LPD) equations can be obtained from this integrable equation by choosing appropriate parameters. The Lax pair and the generalized Darboux transformations of NFONLS are constructed for the first time, from which the \(N\) th order soliton and rational solutions are given in matrix form, and the contour profiles and density evolutions of rational solutions are given to investigate their wave structures and dynamic properties. These results may be useful in nonlinear fiber optics and the relevant physical fields.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
Keywords:
nonlocal nonlinear Schrödinger equation; Lax pair; Darboux transformation; rational solutionReferences:
| [1] | Draper, L., Freak ocean waves, Mar. Obs., 35, 193-195 (1965) |
| [2] | Onorato, M.; Residori, S.; Bortolozzo, U.; Montina, A.; Arecchi, F. T., Rogue wave and their generating mechanisms in different physical contexts, Phys. Rep., 528, 47-89 (2013) |
| [3] | Hennig, H., Taming nonlinear freak waves, Physics, 7, 31 (2014) |
| [4] | Chan, H. N.; Ding, E.; Kedziora, D. J.; Grimshaw, R.; Chow, K. W., Rogue waves for a long wave-short wave resonance model with multiple short waves, Nonlinear Dynam., 85, 2827-2841 (2016) |
| [5] | Yan, Z. Y.; Konotop, V. V.; Akhmediev, N., Three-dimensional rogue waves in non-stationary parabolic potentials, Phys. Rev. E, 82, 036610 (2010) |
| [6] | Solli, D. R.; Ropers, C.; Koonath, P.; Jalali, B., Optical rogue waves, Nature, 450, 1054-1057 (2007) |
| [7] | Akhmediev, N., Roadmap on optical rogue waves and extreme events, J. Opt., 18, 063001 (2016) |
| [8] | Bailung, H.; Sharma, S. K.; Nakamura, Y., Observation of peregrine soliton in a multicomponent plasma with negative ions, Phys. Rev. Lett., 107, 255005 (2011) |
| [9] | Yan, Z. Y., Vector financial rogue waves, Phys. Lett. A, 375, 4274-4279 (2011) · Zbl 1254.91190 |
| [10] | Akhmediev, N.; Ankiewicz, A.; Taki, M., Are rogue waves robust against perturbations?, Phys. Lett. A, 373, 3997-4000 (2009) · Zbl 1234.35237 |
| [11] | Onorato, M., Statistical properties of directional ocean waves: the role of the modulational instability in the formation of extreme events, Phys. Rev. Lett., 102, 114502 (2009) |
| [12] | Kibler, B.; Fatome, J.; Finot, C.; Millot, G.; Dias, F.; Genty, G.; Akhmediev, N.; Dudley, J. M., The Peregrine soliton in nonlinear fibre optics, Nat. Phys., 6, 790-795 (2010) |
| [13] | Ankiewicz, A.; Soto-Crespo, J. M.; Akhmediev, N., Rogue waves and rational solutions of the Hirota equation, Phys. Rev. E, 81, 046602 (2010) |
| [14] | Bandelow, U.; Akhmediev, N., Persistence of rogue waves in extended nonlinear Schrödinger equations: Integrable Sasa-Satsuma case, Phys. Lett. A, 376, 1558-1561 (2012) · Zbl 1260.35195 |
| [15] | Chen, S. H., Twisted rogue-wave pairs in the Sasa-Satsuma equation, Phys. Rev. E, 88, 023202 (2013) |
| [16] | Wang, X.; Yang, B.; Chen, Y.; Yang, Y. Q., Higer-order rogue wave solutions of the Kundu-Eckhaus equation, Phys. Scr., 89, 095210 (2014) |
| [17] | Xie, X. Y.; Tian, B.; Sun, W. R.; Sun, Y., Rogue wave solutions for the Kundu-Eckhaus equation with variable coefficients in an optical fiber, Nonlinear Dynam., 81, 1349-1354 (2015) · Zbl 1348.78020 |
| [18] | Xu, S. W.; He, J. S., The rogue wave and breather solution of the Gerdjikov-Ivanov equation, J. Math. Phys., 53, 063507 (2012) · Zbl 1288.35439 |
| [19] | Wang, L. H.; Porsezian, K.; He, J. S., Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation, Phys. Rev. E, 87, 053202 (2013) |
| [20] | Chowdury, A.; Kedziora, D. J.; Ankiewicz, A.; Akhmediev, N., Breather solutions of the integrable quintic nonlinear Schrödinger equation and their interactions, Phys. Rev. E, 91, 022919 (2015) · Zbl 1374.35381 |
| [21] | Wen, X. Y.; Yang, Y. Q.; Yan, Z. Y., Generalized perturbation \((n, M)\)-fold Darboux transformations and multi-rogue-wave structures for the modified self-steepening nonlinear Schrödinger equation, Phys. Rev. E, 92, 012917 (2015) |
| [22] | Ohta, Y.; Yang, J. K., Rogue waves in the Davey-Stewartson I equation, Phys. Rev. E, 86, 036604 (2012) |
| [23] | Chan, H. N.; Chow, K. W.; Kedziora, D. J.; Grimshaw, R. H.J., Rogue wave modes for a derivative nonlinear Schrödinger model, Phys. Rev. E, 89, 032914 (2014) |
| [24] | Baronio, F.; Degasperis, A.; Conforti, M.; Wabnitz, S., Solutions of the vector nonlinear schrödinger equations: evidence for deterministic Rogue waves, Phys. Rev. Lett., 109, 044102 (2012) |
| [25] | Li, J. H.; Chan, H. N.; Chiang, K. S.; Chow, K. W., Breathers and ‘black’ rogue waves of coupled nonlinear Schrödinger equations with dispersion and nonlinearity of opposite signs, Commun. Nonlinear Sci. Numer. Simul., 28, 28-38 (2015) · Zbl 1510.35306 |
| [26] | Guo, B. L.; Ling, L. M., Rogue wave, breathers and bright-dark-rogue solutions for the coupled schrödinger equations, Chin. Phys. Lett., 28, 110202 (2011) |
| [27] | Dai, C. Q.; Huang, W. H., Multi-rogue wave and multi-breather solutions in PT-symmetric coupled waveguides, Appl. Math. Lett., 32, 35-40 (2014) · Zbl 1311.35276 |
| [28] | Ablowitz, M. J.; Prinari, B.; Trubatch, A. D., Discrete and Continuous Nonlinear SchrÖDinger Systems (2004), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1057.35058 |
| [29] | Dudley, J. M.; Taylor, J. R., Supercontinuum Generation in Optical Fibers (2010), Cambridge University Press: Cambridge University Press Cambridge |
| [30] | Kodama, Y.; Hasegawa, A., Nonlinear pulse propagation in a monomode dielectric guide, IEEE J. Quantum Electron., 23, 510-515 (1987) |
| [31] | Lakshmanan, M.; Porsezian, K.; Daniel, M., Effect on discreteness on the continuum limit of the Heisenberg spin chain, Phys. Lett. A, 133, 483-488 (1988) |
| [32] | Kano, T., Normal Form of Nonlinear Schrödinger Equation, J. Phys. Soc. Japan, 58, 4322-4328 (1989) |
| [33] | Dai, C. Q.; Wang, X. G.; Zhou, G. Q., Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials, Phys. Rev. A, 89, 013834 (2014) |
| [34] | Dai, C. Q.; Fan, Y.; Zhou, G. Q.; Zheng, J.; Chen, L., Vector spatiotemporal localized structures in \((3 + 1)\)-dimensional strongly nonlocal nonlinear media, Nonlinear Dynam., 86, 999-1005 (2016) |
| [35] | Musslimani, Z. H., Optical Solitons in PT Periodic Potentials, Phys. Rev. Lett., 100, 030402 (2008) |
| [36] | Li, J. H.; Ren, H. D.; Pei, S. X.; Cao, Z. L.; Xian, F. L., Modulation instabilities in randomly birefringent two-mode optical fibers, Chin. Phys. B, 25, 124208 (2016) |
| [37] | Ablowitz, M. J.; Musslimani, Z. H., Integrable nonlocal nonlinear Schrödinger equation, Phys. Rev. Lett., 110, 064105 (2013) |
| [38] | Ablowitz, M. J.; Musslimani, Z. H., Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation, Nonlinearity, 29, 915 (2016) · Zbl 1338.37099 |
| [39] | Gadzhimuradov, T. A.; Agalarov, A. M., Towards a gauge-equivalent magnetic structure of the the nonlocal nonlinear Schrödingger equation, Phys. Rev. A, 93, 062124 (2016) |
| [40] | Rybakov, F. N.; Borisov A. B, Blügel, S.; Kiselev, N. S., New type of stable particlelike states in chiral Magnets, Phys. Rev. Lett., 115, 117201 (2015) |
| [41] | Dantas, C. C., An inhomogeneous space-time patching model based on a nonlocal and nonlinear Schrödiner equation, Found. Phys., 46, 1269-1292 (2016) · Zbl 1368.83079 |
| [42] | Ablowitz, M. J.; Musslimani, Z. H., Integrable nonlocal nonlinear qquations, Stud. Appl. Math., 139, 7 (2017) · Zbl 1373.35281 |
| [43] | Li, M.; Xu, T., Dark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential, Phys. Rev. E, 91, 033202 (2015) |
| [44] | Wen, X. Y.; Yan, Z. Y.; Yang, Y. Q., Dynamics of higer-order rational solitons for the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential, Chaos, 26, 063123 (2016) |
| [45] | Xu, Z. X.; Chow, K. W., Breathers and rogue waves for a third order nonlocal partial differential equation by a bilinear transformation, Appl. Math. Lett., 56, 72-77 (2016) · Zbl 1339.35081 |
| [46] | Liu, W.; Qiu, D. Q.; Wu, Z. W.; He, J. S., Dynamical behabior of solutions in integrable nonlocal Lakshmanan-Porsezian-Daniel equation, Commun. Theor. Phys., 65, 671-676 (2016) · Zbl 1345.35104 |
| [47] | Yang, Y. Q.; Yan, Z. Y.; Malomed, B. A., Rogue waves, rational solitons, and modulational instability in an integrable fifth-order nonlinear Schrödinger equation, Chaos, 25, 103112 (2015) · Zbl 1374.35392 |
| [48] | Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0762.35001 |
| [49] | Rogers, C.; Shadwick, W. F., Bäcklund Transformations and their Applications (1982), Academic Press: Academic Press New York · Zbl 0492.58002 |
| [50] | Matveev, V. B.; Salle, M. A., Darboux Transformation and Solitons (1991), Springer-Verlag Press: Springer-Verlag Press Berlin · Zbl 0744.35045 |
| [51] | Hirota, R., The Direct Methods in Soliton Theory (2004), Cambridge University Press: Cambridge University Press Cambridge |
| [52] | Deift, P.; Trubowitz, E., Inverse scattering on the line, Comm. Pure Appl. Math., 32, 121-251 (1979) · Zbl 0388.34005 |
| [53] | Kong, L. Q.; Dai, C. Q., Some discussions about variable separation of nonlinear models using Riccati equation expansion method, Nonlinear Dynam., 81, 1553-1561 (2015) |
| [54] | Wazwaz, A. M., Multiple soliton solutions and multiple complex soliton solutions for two distinct Boussinesq equations, Nonlinear Dynam., 85, 731-737 (2016) |
| [55] | Dai, C. Q., Controllable combined Peregrine soliton and Kuznetsov-Ma soliton in PT-symmetric nonlinear couplers with gain and loss, Nonlinear Dynam., 80, 715-721 (2015) |
| [56] | Akhmediev, N.; Ankiewicz, A.; Soto-Crespo, J. M., Rogue waves and rational solutions of the nonlinear Schrödinger equation, Phys. Rev. E, 80, 026601 (2009) |
| [57] | Guo, B. L.; Ling, L. M.; Liu, Q. P., Nonlinear Schrödiger equation: Generalzied Darboux transformation and rogue wave solutions, Phys. Rev. E, 85, 026607 (2012) |
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