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Higher-order vector discrete rogue-wave states in the coupled Ablowitz-Ladik equations: exact solutions and stability. (English) Zbl 1378.35284

Summary: An integrable system of two-component nonlinear Ablowitz-Ladik equations is used to construct complex rogue-wave (RW) solutions in an explicit form. First, the modulational instability of continuous waves is studied in the system. Then, new higher-order discrete two-component RW solutions of the system are found by means of a newly derived discrete version of a generalized Darboux transformation. Finally, the perturbed evolution of these RW states is explored in terms of systematic simulations, which demonstrates that tightly and loosely bound RWs are, respectively, nearly stable and strongly unstable solutions.{
©2016 American Institute of Physics}

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35C05 Solutions to PDEs in closed form
35A22 Transform methods (e.g., integral transforms) applied to PDEs
35B35 Stability in context of PDEs
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References:

[1] Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering, (1990), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 0762.35001
[2] Pitaevskii, L.; Stringari, S., Bose-Einstein Condensation, (2003), Oxford University Press: Oxford University Press, Oxford · Zbl 1110.82002
[3] Bagnato, V. S.; Frantzeskakis, D. J.; Kevrekidis, P. G.; Malomed, B. A.; Mihalache, D., Rom. Rep. Phys., 67, 5, (2015)
[4] Dudley, J. M.; Dias, F.; Erkintalo, M.; Genty, G., Nat. Phys., 8, 755, (2014)
[5] Akhmediev, N.; Kibler, B.; Baronio, F.; Belić, M.; Zhong, W.-P.; Zhang, Y.; Chang, W.; Soto-Crespo, J. M.; Vouzas, P.; Grelu, P.; Lecaplain, C.; Hammani, K.; Rica, S.; Picozzi, A.; Tlidi, M.; Panajotov, K.; Mussot, A.; Bendahmane, A.; Szriftgiser, P.; Genty, G.; Dudley, J.; Kudlinski, A.; Demircan, A.; Morgner, U.; Amiramashvili, S.; Bree, C.; Steinmeyer, G.; Masoller, C.; Broderick, N. G. R.; Runge, A. F. J.; Erkintalo, M.; Residori, S.; Bortolozzo, U.; Arecchi, F. T.; Wabnitz, S.; Tiofack, C. G.; Coulibaly, S.; Taki, M., J. Opt., 18, 063001, (2016)
[6] Malomed, B. A.; Scott, A., Nonlinear Schrödinger equations, Encyclopedia of Nonlinear Science, 639, (2005), Routledge: Routledge, New York
[7] Malomed, B. A.; Mihalache, D.; Wise, F.; Torner, L., J. Opt. B, 7, R53-R72, (2005), 10.1088/1464-4266/7/5/R02; Malomed, B. A.; Mihalache, D.; Wise, F.; Torner, L., J. Opt. B, 7, R53-R72, (2005), 10.1088/0953-4075/49/17/170502;
[8] Carretero-González, R.; Frantzeskakis, D. J.; Kevrekidis, P. G., Nonlinearity, 21, R139, (2008) · Zbl 1216.82023
[9] Kartashov, Y. V.; Malomed, B. A.; Torner, L., Rev. Mod. Phys., 83, 247, (2011)
[10] Ma, W. X.; Chen, M., Appl. Math. Comput., 215, 2835, (2009)
[11] Ivancevic, V., Cognit. Comput., 2, 17, (2010)
[12] Yan, Z., Commun. Theor. Phys., 54, 947, (2010) · Zbl 1219.91143
[13] Yan, Z., Phys. Lett. A, 375, 4274, (2011) · Zbl 1254.91190
[14] Akhmediev, N.; Soto-Crespo, J. M.; Ankiewicz, A., Phys. Rev. A, 80, 043818, (2009)
[15] Ankiewicz, A.; Wang, Y.; Wabnitz, S.; Akhmediev, N., Phys. Rev. E, 89, 012907, (2014)
[16] Yan, Z.; Konotop, V. V.; Akhmediev, N., Phys. Rev. E, 82, 036610, (2010)
[17] Yan, Z., Phys. Lett. A, 374, 672, (2010) · Zbl 1235.35266
[18] Guo, B. L.; Ling, L. M.; Liu, Q. P., Phys. Rev. E, 85, 026607, (2012)
[19] Chen, S.; Mihalache, D., J. Phys. A, 48, 215202, (2015) · Zbl 1317.35217
[20] Ohta, Y.; Yang, J., Phys. Rev. E, 86, 036604, (2012)
[21] Wen, X.; Yang, Y.; Yan, Z., Phys. Rev. E, 92, 012917, (2015)
[22] Wen, X.; Yan, Z., Chaos, 25, 123115, (2015) · Zbl 1374.37092
[23] Yang, Y.; Yan, Z.; Malomed, B. A., Chaos, 25, 103112, (2015) · Zbl 1374.35392
[24] Yan, Z., Nonlinear Dyn., 79, 2515, (2015)
[25] Wen, X. Y.; Yan, Z.; Yang, Y., Chaos, 26, 063123, (2016)
[26] Chen, S.; Soto-Crespo, J. M.; Baronio, F.; Grelu, Ph.; Mihalache, D., Opt. Express, 24, 15251, (2016)
[27] Yuan, F.; Rao, J.; Porsezian, K.; Mihalache, D.; He, J. S., Rom. J. Phys., 61, 378, (2016)
[28] Ablowitz, M. J.; Ladik, J. F., J. Math. Phys., 16, 598, (1975) · Zbl 0296.34062
[29] Ankiewicz, A.; Akhmediev, N.; Soto-Crespo, J. M., Phys. Rev. E, 82, 026602, (2010)
[30] Ankiewicz, A.; Akhmediev, N.; Lederer, F., Phys. Rev. E, 83, 056602, (2011)
[31] Ohta, Y.; Yang, J., J. Phys. A: Math. Theor., 47, 255201, (2014) · Zbl 1294.35121
[32] Wen, X. Y. and Yan, Z., “Modulational instability and dynamical behaviors of discrete multi-rogue waves of the Ablowitz-Ladik equation” (unpublished). · Zbl 1414.35212
[33] Szameit, A.; Dreisow, F.; Heinrich, M.; Nolte, S.; Sukhorukov, A. A., Phys. Rev. Lett., 106, 193903, (2011)
[34] Dutta, O.; Gajda, M.; Hauke, P.; Lewenstein, M.; Luhmann, D.-S.; Malomed, B.; Sowinski, T.; Zakrzewski, J., Rep. Prog. Phys., 78, 066001, (2015)
[35] Malomed, B. A.; Yang, J., Phys. Lett. A, 302, 163, (2002) · Zbl 0998.37022
[36] Wen, X. Y.; Wang, D. S.; Meng, X. H., Rep. Math. Phys., 72, 349, (2013) · Zbl 1396.37076
[37] Ankiewicz, A.; Devine, N.; Ünal, M.; Chowdury, A.; Akhmediev, N., J. Opt., 15, 064008, (2013)
[38] Wen, X. Y., J. Phys. Soc. Jpn., 81, 114006, (2012)
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