Analytical solutions for the coupled Hirota equations in the firebringent fiber. (English) Zbl 07426878

Summary: Under investigation in this paper are the coupled Hirota (CH) equations, which describe the collision of two waves in the deep ocean and the propagation of the ultrashort optical pulses in a birefringent fiber. Based on the bilinear method, multi-soliton solutions for the CH equations are given. From the perspective of analysis, the interaction dynamics of solitons are obtained. The head-on and overtaking interactions of two/three solitons are analyzed. The elastic and inelastic interactions of two/three solitons are presented. The soliton velocity can be controlled by adjusting the physical parameters \(\phi_j\), \((j = 1,2, \dots N )\) and \(\epsilon\). The energy exchange occurs between the variables \(u\) and \(v\) before and after the collision. The local interference of two/three solitons is observed. The closer to the center of the interaction, the more significant the interference. The real and imaginary parts of the parameters \(\phi_j\), \((j = 1,2,\dots N )\) have the effects on the numbers of peaks and holes in the collisions. The structure of four eyes and one peak is observed during the two-soliton interaction. The three-soliton interactions are given with two peaks and local interference. It is hoped that the results of this study can provide some reference for the wave interaction in deep ocean, pulse propagation in optical fiber, financial option pricing, valuation of intangible assets in sports and so on.


35Qxx Partial differential equations of mathematical physics and other areas of application
78Axx General topics in optics and electromagnetic theory
37Kxx Dynamical system aspects of infinite-dimensional Hamiltonian and Lagrangian systems
Full Text: DOI


[1] Gao, X. Y.; Guo, Y. J.; Shan, W. R., Shallow water in an open sea or a wide channel: auto- and non-auto-Bäcklund transformations with solitons for a generalized (2 + 1)-dimensional dispersive long-wave system, Chaos Solitons Fractals, 138, 109950 (2020)
[2] Gao, X. Y.; Guo, Y. J.; Shan, W. R., Water-wave symbolic computation for the Earth, Enceladus and Titan: the higher-order Boussinesq-Burgers system, auto- and non-auto-Bäcklund transformations, Appl. Math. Lett., 104, 106170 (2020) · Zbl 1437.86001
[3] M.J. Ablowitz, P.A. Clarkson, 1991, Cambridge Univ. Press, Cambridge
[4] Gibbon, J. D.; Radmore, P.; Tabor, M.; Wood, D., The Painleve property and Hirotas method, Stud. Appl. Math., 72, 39-63 (1985) · Zbl 0581.35074
[5] Tamura, K.; Nelson, L. E.; Haus, H. A.; Ippen, E. P., Soliton versus nonsoliton operation of fiber ring lasers, Appl. Phys. Lett., 64, 149-151 (1994)
[6] Zhu, S. H.; Gao, Y. T.; Yu, X.; Sun, Z. Y.; Gai, X. L.; Meng, D. X., Painleve property, soliton-like solutions and complexitons for a coupled variable-coefficient modified Korteweg-de Vries system in a two-layer fluid model, Appl. Math. Comput., 217, 295-307 (2010) · Zbl 1277.35310
[7] Guo, B. L.; Ling, L. M.; Liu, Q. P., Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions, Phys. Rev. E, 85, 26607 (2012)
[8] Gai, X. L.; Gao, Y. T.; Xin, Y.; Lei, W., Painleve property, auto-Bäcklund transformation and analytic solutions of a variable-coefficient modified Korteweg-de Vries model in a hot magnetized dusty plasma with charge fluctuations, Appl. Math. Comput., 218, 271-279 (2011) · Zbl 1275.76227
[9] Christov, I.; Christov, C. I., Physical dynamics of quasi-particles in nonlinear wave equations, Phys. Lett. A, 372, 841-848 (2008) · Zbl 1217.81052
[10] Mo, J. Q., Approximation of the soliton solution for the generalized Vakhnenko equation, Chin. Phys. B, 18, 4608 (2009)
[11] Nowak, X.; Söffker, D., Data-driven stabilization of unknown nonlinear dynamical systems using a cognition-based framework, Nonlinear Dyn., 86, 1-15 (2016) · Zbl 1349.93306
[12] Gao, X. Y.; Guo, Y. J.; Shan, W. R.; Yuan, Y. Q.; Zhang, C. R.; Chen, S. S., Magneto-optical/ferromagnetic-material computation: Bäcklund transformations, bilinear forms and n solitons for a generalized (3+1)-dimensional variable-coefficient modified kadomtsev-petviashvili system, Appl. Math. Lett., 111, 106627 (2021) · Zbl 1455.35248
[13] Du, X. X.; Tian, B.; Qu, Q. X.; Yuan, Y. Q.; Zhao, X. H., Lie group analysis, solitons, self-adjointness and conservation laws of the modified Zakharov-Kuznetsov equation in an electron-positron-ion magnetoplasma, Chaos Solitons Fractals, 134, 109709 (2020)
[14] Wang, M.; Tian, B.; Sun, Y.; Zhang, Z., Lump, mixed lump-stripe and rogue wave-stripe solutions of a (3 + 1)-dimensional nonlinear wave equation for a liquid with gas bubbles, Comput. Math. Appl., 79, 576-587 (2020) · Zbl 1443.76233
[15] Zhao, X.; Tian, B.; Qu, Q. X.; Yuan, Y. Q.; Du, X. X.; Chu, M. X., Dark-dark solitons for the coupled spatially modulated Gross-Pitaevskii system in the Bose-Einstein condensation, Mod. Phys. Lett. B, 34, 2050282 (2020)
[16] Chen, Y. Q.; Tian, B.; Qu, Q. X.; Li, H.; Zhao, X. H.; Tian, H. Y.; Wang, M., Ablowitz-Kaup-Newell-Segzur system, conservation laws and Bäcklund transformation of a variable-coefficient Korteweg-de Vries equation in plasma physics, fluid dynamics or atmospheric science, Int. J. Mod. Phys. B, 34, 2050226 (2020) · Zbl 1451.35158
[17] Gao, X. Y.; Guo, Y. J.; Shan, W. R., Oceanic studies via a variable-coefficient nonlinear dispersive-wave system in the Solar system, Chaos Solitons Fractals, 142, 110367 (2021)
[18] Weiss; John, Modified equations, rational solutions, and the Painleve property for the Kadomtsev-Petviashvili and Hirota-Satsuma equations, J. Math. Phys., 26, 2174 (1985) · Zbl 0588.35020
[19] Porsezian, K.; Nakkeeran, K., Optical solitons in birefringent fibre-Bäcklund transformation approach, Pure Appl. Opt., 6, L7 (1997)
[20] Ankiewicz, A.; Soto-Crespo, J. M.; Akmediev, N., Rogue waves and rational solutions of the Hirota equation, Phys. Rev. E, 81, 046602 (2010)
[21] Chen, S. H.; Song, L. Y., Rogue waves in coupled Hirota systems, Phys. Rev. E, 87, 032910 (2013)
[22] Tasgal, R. S.; Potasek, M. J., Soliton solutions to coupled higher-order nonlinear Schrödinger equations, J. Math. Phys., 33, 1208 (1992)
[23] Bindu, S. G.; Mahalingam, A.; Porsezian, K., Dark solitons of the coupled Hirota equaiton in nonlinear fiber, Phys. Lett. A, 286, 321-331 (2001) · Zbl 0971.78019
[24] Xie, X. Y.; Liu, X. B., Elastic and inelastic collisions of the semirational solutions for the coupled Hirota equations in a birefingent fiber, Appl. Math. Lett., 105, 106291 (2020) · Zbl 1437.35642
[25] Hirota, R., Exact solution of the Korteweg-de Vries equation formultip le collisions of solitons, Phys. Rev. Lett., 27, 1192-1194 (1971) · Zbl 1168.35423
[26] Radhakrishnan, R.; Sahadevan, R.; Lakshmanan, M., Integrability and singularity structure of coupled nonlinear Schrödinger equations, Chaos Solitons Fractals, 5, 2315-2327 (1995) · Zbl 1080.35546
[27] Porsezain, K.; Nakkeeran, K., Optical solitons in presence of kerr dispersion and self-frequency shift, Phys. Rev. Lett., 76, 3955 (1996)
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