×

Baseband modulation instability, rogue waves and state transitions in a deformed Fokas-Lenells equation. (English) Zbl 1430.37077

Summary: We study a deformed Fokas-Lenells equation which is related to the integrable derivative nonlinear Schrödinger hierarchy with higher-order nonholonomic constraint. The baseband modulation instability as an origin of rogue waves is displayed. The explicit rogue wave solutions are obtained via the Darboux transformation. Typical rogue wave patterns such as the standard rogue wave, dark rogue wave and twisted rogue wave pair in three different components of the deformed Fokas-Lenells equation are presented. Besides, the state transitions between rogue waves and solitons are analytically found when the modulation instability growth rate tends to zero in the zero-frequency perturbation region. The explicit soliton solutions under the special parameter condition are given. The anti-dark and W-shaped solitons in their respective components are shown.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q55 NLS equations (nonlinear Schrödinger equations)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35C08 Soliton solutions
35Q51 Soliton equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Fokas, A.S.: On a class of physically important integrable equations. Physica D 87, 145 (1995) · Zbl 1194.35363 · doi:10.1016/0167-2789(95)00133-O
[2] Lenells, J.: Exactly solvable model for nonlinear pulse propagation in optical fibers. Stud. Appl. Math. 123, 215 (2009) · Zbl 1171.35473 · doi:10.1111/j.1467-9590.2009.00454.x
[3] Lenells, J., Fokas, A.S.: On a novel integrable generalization of the nonlinear Schrödinger equation. Nonlinearity 22, 11 (2008) · Zbl 1160.35536 · doi:10.1088/0951-7715/22/1/002
[4] Lenells, J., Fokas, A.S.: An integrable generalization of the nonlinear Schrödinger equation on the half-line and solitons. Inverse Probl. 25, 115006 (2009) · Zbl 1181.35335 · doi:10.1088/0266-5611/25/11/115006
[5] Lenells, J.: Dressing for a novel integrable generalization of the nonlinear Schrödinger equation. J. Nonlinear Sci. 20, 709 (2010) · Zbl 1209.35130 · doi:10.1007/s00332-010-9070-1
[6] Matsuno, Y.: A direct method of solution for the Fokas-Lenells derivative nonlinear Schrödinger equation: II. Dark soliton solutions. J. Phys. A Math. Theor. 45, 475202 (2012) · Zbl 1253.82068 · doi:10.1088/1751-8113/45/47/475202
[7] Wright III, O.C.: Some homoclinic connections of a novel integrable generalized nonlinear Schrödinger equation. Nonlinearity 22, 2633 (2009) · Zbl 1179.35315 · doi:10.1088/0951-7715/22/11/003
[8] He, J.S., Xu, S.W., Porsezian, K.: Rogue waves of the Fokas-Lenells equation. J. Phys. Soc. Jpn. 81, 124007 (2012) · doi:10.1143/JPSJ.81.124007
[9] Chen, S.H., Song, L.Y.: Peregrine solitons and algebraic soliton pairs in Kerr media considering spacetime correction. Phys. Lett. A 378, 1228 (2014) · Zbl 1331.37103 · doi:10.1016/j.physleta.2014.02.042
[10] Triki, H., Wazwaz, A.M.: Combined optical solitary waves of the Fokas-Lenells equation. Wave Random Complex 27, 587 (2017) · Zbl 07659360 · doi:10.1080/17455030.2017.1285449
[11] Guo, B.L., Ling, L.M.: Riemann-Hilbert approach and N-soliton formula for coupled derivative Schrödinger equation. J. Math. Phys. 53, 073506 (2012) · Zbl 1276.81068 · doi:10.1063/1.4732464
[12] Zhang, Y., Yang, J.W., Chow, K.W., Wu, C.F.: Solitons, breathers and rogue waves for the coupled Fokas-Lenells system via Darboux transformation. Nonlinear Anal. RWA 33, 237 (2017) · Zbl 1352.35170 · doi:10.1016/j.nonrwa.2016.06.006
[13] Chen, S.H., Ye, Y., Soto-Crespo, J.M., Grelu, P., Baronio, F.: Peregrine solitons beyond the threefold limit and their two-soliton interactions. Phys. Rev. Lett. 121, 104101 (2018) · doi:10.1103/PhysRevLett.121.104101
[14] Ling, L.M., Feng, B.F., Zhu, Z.N.: General soliton solutions to a coupled Fokas-Lenells equation. Nonlinear Anal. RWA 40, 185 (2018) · Zbl 1382.35068 · doi:10.1016/j.nonrwa.2017.08.013
[15] Kupershmidt, B.A.: KdV6: an integrable system. Phys. Lett. A 372, 2634 (2008) · Zbl 1220.35153 · doi:10.1016/j.physleta.2007.12.019
[16] Kundu, A.: Integrable twofold hierarchy of perturbed equations and application to optical soliton dynamics. Theor. Math. Phys. 167, 800 (2011) · doi:10.1007/s11232-011-0063-5
[17] Wang, X., Liu, C., Wang, L.: Rogue waves and W-shaped solitons in the multiple self-induced transparency system. Chaos 27, 093106 (2017) · Zbl 1390.35047 · doi:10.1063/1.4986609
[18] Ren, Y., Yang, Z.Y., Liu, C., Yang, W.L.: Different types of nonlinear localized and periodic waves in an erbium-doped fiber system. Phys. Lett. A 379, 2991 (2015) · doi:10.1016/j.physleta.2015.08.037
[19] Wang, L., Liu, C., Wu, X., Wang, X., Sun, W.R.: Dynamics of superregular breathers in the quintic nonlinear Schrödinger equation. Nonlinear Dyn. 94, 977 (2018) · doi:10.1007/s11071-018-4404-x
[20] Wang, L., Wu, X., Zhang, H.Y.: Superregular breathers and state transitions in a resonant erbium-doped fiber system with higher-order effects. Phys. Lett. A 382, 2650 (2018) · doi:10.1016/j.physleta.2018.07.036
[21] Ren, Y., Liu, C., Yang, Z.Y., Yang, W.L.: Polariton superregular breathers in a resonant erbium-doped fiber. Phys. Rev. E 98, 062223 (2018) · doi:10.1103/PhysRevE.98.062223
[22] Kundu, A.: Two-fold integrable hierarchy of nonholonomic deformation of the derivative nonlinear Schrödinger and the Lenells-Fokas equation. J. Math. Phys. 51, 022901 (2010) · Zbl 1309.37060 · doi:10.1063/1.3276447
[23] Solli, D.R., Ropers, C., Koonath, P., Jalali, B.: Optical rogue waves. Nature 450, 1054 (2007) · doi:10.1038/nature06402
[24] Dysthe, K., Krogstad, H.E., Muller, P.: Oceanic rogue waves. Annu. Rev. Fluid Mech. 40, 287 (2008) · Zbl 1136.76009 · doi:10.1146/annurev.fluid.40.111406.102203
[25] Akhmediev, N., Ankiewicz, A., Soto-Crespo, J.M.: Rogue waves and rational solutions of the nonlinear Schrödinger equation. Phys. Rev. E 80, 026601 (2009) · doi:10.1103/PhysRevE.80.026601
[26] Akhmediev, N., Ankiewicz, A., Taki, M.: Waves that appear from nowhere and disappear without a trace. Phys. Lett. A 373, 675 (2009) · Zbl 1227.76010 · doi:10.1016/j.physleta.2008.12.036
[27] Peregrine, D.H.: Water waves, nonlinear Schrödinger equations and their solutions. J. Aust. Math. Soc. Ser. B Appl. Math. 25, 16 (1983) · Zbl 0526.76018 · doi:10.1017/S0334270000003891
[28] Guo, B.L., Ling, L.M., Liu, Q.P.: Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions. Phys. Rev. E 85, 026607 (2012) · doi:10.1103/PhysRevE.85.026607
[29] He, J.S., Zhang, H.R., Wang, L.H., Porsezian, K., Fokas, A.S.: Generating mechanism for higher-order rogue waves. Phys. Rev. E 87, 052914 (2013) · doi:10.1103/PhysRevE.87.052914
[30] Zhang, G.Q., Yan, Z.Y., Wen, X.Y., Chen, Y.: Interactions of localized wave structures and dynamics in the defocusing coupled nonlinear Schrödinger equations. Phys. Rev. E 95, 042201 (2017) · doi:10.1103/PhysRevE.95.042201
[31] Wei, J., Wang, X., Geng, X.G.: Periodic and rational solutions of the reduced Maxwell-Bloch equations. Commun. Nonlinear Sci. Numer. Simul. 59, 1 (2018) · Zbl 1510.78011 · doi:10.1016/j.cnsns.2017.10.017
[32] Wang, X., Zhang, J.L., Wang, L.: Conservation laws, periodic and rational solutions for an extended modified Korteweg – de Vries equation. Nonlinear Dyn. 92, 1507 (2018) · doi:10.1007/s11071-018-4143-z
[33] Li, P., Wang, L., Kong, L.Q., Wang, X., Xie, Z.Y.: Nonlinear waves in the modulation instability regime for the fifth-order nonlinear Schrödinger equation. Appl. Math. Lett. 85, 110 (2018) · Zbl 1405.35197 · doi:10.1016/j.aml.2018.05.027
[34] Liu, J.G., Zhang, Y.F.: Construction of lump soliton and mixed lump stripe solutions of (3+1)-dimensional soliton equation. Results Phys. 10, 94 (2018) · doi:10.1016/j.rinp.2018.05.022
[35] Liu, J.G., Zhang, Y.F., Muhammad, I.: Resonant soliton and complexiton solutions for (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Comput. Math. Appl. 75, 3939 (2018) · Zbl 1420.35321 · doi:10.1016/j.camwa.2018.03.004
[36] Liu, J.G., Yang, X.J., Cheng, M.H., Feng, Y.Y., Wang, Y.D.: Abound rogue wave type solutions to the extended (3+ 1)-dimensional Jimbo-Miwa equation. Comput. Math. Appl. (2019). https://doi.org/10.1016/j.camwa.2019.03.034 · Zbl 1442.35387
[37] Chen, J.C., Zhu, S.D.: Residual symmetries and soliton – cnoidal wave interaction solutions for the negative-order Korteweg – de Vries equation. Appl. Math. Lett. 73, 136 (2017) · Zbl 1375.35018 · doi:10.1016/j.aml.2017.05.002
[38] Chen, J.C., Ma, Z.Y.: Consistent Riccati expansion solvability and soliton – cnoidal wave interaction solution of a (2 + 1)-dimensional Korteweg – de Vries equation. Appl. Math. Lett. 64, 87 (2017) · Zbl 1354.35128 · doi:10.1016/j.aml.2016.08.016
[39] Chen, J.C., Ma, Z.Y., Hu, Y.H.: Nonlocal symmetry, Darboux transformation and soliton – cnoidal wave interaction solution for the shallow water wave equation. J. Math. Anal. Appl. 460, 987 (2018) · Zbl 1383.35009 · doi:10.1016/j.jmaa.2017.12.028
[40] Ankiewicz, A., Soto-Crespo, J.M., Akhmediev, N.: Rogue waves and rational solutions of the Hirota equation. Phys. Rev. E 81, 046602 (2010) · doi:10.1103/PhysRevE.81.046602
[41] Tao, Y.S., He, J.S.: Multisolitons, breathers, and rogue waves for the Hirota equation generated by the Darboux transformation. Phys. Rev. E 85, 026601 (2012) · doi:10.1103/PhysRevE.85.026601
[42] Bandelow, U., Akhmediev, N.: Persistence of rogue waves in extended nonlinear Schrödinger equations: integrable Sasa-Satsuma case. Phys. Lett. A 376, 1558 (2012) · Zbl 1260.35195 · doi:10.1016/j.physleta.2012.03.032
[43] Chen, S.H.: Twisted rogue-wave pairs in the Sasa-Satsuma equation. Phys. Rev. E 88, 023202 (2013) · doi:10.1103/PhysRevE.88.023202
[44] 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) · doi:10.1103/PhysRevLett.109.044102
[45] Ling, L.M., Guo, B.L., Zhao, L.C.: High-order rogue waves in vector nonlinear Schrödinger equations. Phys. Rev. E 89, 041201 (2014) · doi:10.1103/PhysRevE.89.041201
[46] Chen, S.H., Song, L.Y.: Rogue waves in coupled Hirota systems. Phys. Rev. E 87, 032910 (2013) · doi:10.1103/PhysRevE.87.032910
[47] Wang, X., Liu, C., Wang, L.: Darboux transformation and rogue wave solutions for the variable-coefficients coupled Hirota equations. J. Math. Anal. Appl. 449, 1534 (2017) · Zbl 1361.35160 · doi:10.1016/j.jmaa.2016.12.079
[48] Baronio, F., Conforti, M., Degasperis, A., Lombardo, S., Onorato, M., Wabnitz, S.: Vector rogue waves and baseband modulation instability in the defocusing regime. Phys. Rev. Lett. 113, 034101 (2014) · doi:10.1103/PhysRevLett.113.034101
[49] Chen, S., Baronio, F., Soto-Crespo, J.M., Grelu, P., Mihalache, D.: Versatile rogue waves in scalar, vector, and multidimensional nonlinear systems. J. Phys. A Math. Theor. 50, 463001 (2017) · Zbl 1386.76097 · doi:10.1088/1751-8121/aa8f00
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.