The main objective of this work is to study the multi-soliton solutions for the higher-order coupled nonlinear Schrödinger system in an optical fiber. Firstly, by using the Riemann–Hilbert (RH) approach as well as analyzing the related spectral problem of the Lax pair, a matrix RH problem for the system is strictly formulated. Secondly, a series of multi-soliton solutions including breathers and interaction solutions can be computed from the RH problem with the reflectionless case. Thirdly, the propagation and collision dynamic behaviors as well as localized wave characteristics of these solutions are presented by selecting appropriate parameters with some graphics. The innovation and highlights of this article are shown through obtained interesting results. The one is that the higher-order linear and nonlinear term \(\varepsilon \) has important impact on the velocity, phase, period, and wavewidth of wave dynamics. The other is that collisions for the second-order breathers and soliton solutions are elastic interaction which imply they remain bounded all the time. Nevertheless, third-order breathers and soliton solutions are inelastic interaction and the amplitude decreases rapidly with time when collisions occur.
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Wazwaz, A.M., Kaur, L.: New integrable Boussinesq equations of distinct dimensions with diverse variety of soliton solutions. Nonlinear Dyn. 97, 83–94 (2019)
Xu, G.Q., Wazwaz, A.M.: Bidirectional solitons and interaction solutions for a new integrable fifth-order nonlinear equation with temporal and spatial dispersion. Nonlinear Dyn. 101, 581–595 (2020)
Wazwaz, A.M.: Two new integrable fourth-order nonlinear equations: multiple soliton solutions and multiple complex soliton solutions. Nonlinear Dyn. 94, 2655–2663 (2018)
Xu, G.Q., Wazwaz, A.M.: Integrability aspects and localized wave solutions for a new (4+1)-dimensional Boiti–Leon–Manna–Pempinelli equation. Nonlinear Dyn. 98, 1379–1390 (2019)
Wazwaz, A.M., Xu, G.Q.: Kadomtsev–Petviashvili hierarchy: two integrable equations with time-dependent coefficients. Nonlinear Dyn. 100, 3711–3716 (2020)
Gupta, S.C.: Textbook on Optical Fiber Communication and Its Applications. Prentice Hall of India, Delhi (2018)
Yang, J.K.: Nonlinear Waves in Integrable and Nonintegrable Systems. SIAM, Philadelphia (2010)
Ablowitz, M.J.: Nonlinear Dispersive Waves. Cambridge University Press, Cambridge (2011)
Chakraborty, S., Nandy, S., Barthakur, A.: Bilinearization of the generalized coupled nonlinear Schrödinger equation with variable coefficients and gain and dark-bright pair soliton solutions. Phys. Rev. E 91, 023210 (2015)
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)
Wang, D.S., Yin, S.J., Ye, T., Liu, Y.F.: Integrability and bright soliton solutions to the coupled nonlinear Schrödinger equation with higher-order effects. Appl. Math. Comput. 229, 296–309 (2014)
Sun, W.R., Liu, D.Y., Xie, X.Y.: Vector semirational rogue waves and modulation instability for the coupled higher-order nonlinear Schrödinger equations in the birefringent optical fibers. Chaos 27, 043114 (2017)
Akhmediev, N., Soto-Crespo, J.M., Devine, N.: Breather turbulence versus soliton turbulence: Rogue waves, probability density functions, and spectral features. Phys. Rev. E 94, 022212 (2016)
Hasegawa, A., Tappert, F.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Appl. Phys. Lett. 23, 142–144 (1973)
Liu, L., Tian, B., Yuan, Y.Q., Du, Z.: Dark-bright solitons and semirational rogue waves for the coupled Sasa–Satsuma equations. Phys. Rev. E 97, 052217 (2018)
Chai, H.P., Tian, B., Du, Z.: Localized waves for the mixed coupled Hirota equations in an optical fiber. Commun. Nonlin. Sci. Numer. Simulat. 70, 181–192 (2019)
Ding, C.C., Gao, Y.T., Su, J.J., Deng, G.F., Jia, S.L.: Vector semirational rogue waves for the coupled nonlinear Schrödinger equations with the higherorder effects in the elliptically birefringent optical fiber. Wave Random Complex (2018). https://doi.org/10.1080/17455030.2018.1483092
Zhang, G., Yan, Z., Wang, L.: The general coupled hirota equations: modulational instability and higher-order vector rogue wave and multi-dark soliton structures. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 475, 20180625 (2019)
Chen, S.S., Tian, B., Liu, L., Yuan, Y.Q., Zhang, C.R.: Conservation laws, binary Darboux transformations and solitons for a higher-order nonlinear Schrödinger system. Chaos Solitons Fractals 118, 337–346 (2019)
Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)
Yu, F.J.: Inverse scattering solutions and dynamics for a nonlocal nonlinear Gross–Pitaevskii equation with PT-symmetric external potentials. Appl. Math. Lett. 92, 108–114 (2019)
Guo, H.D., Xia, T.C., Hu, B.B.: Dynamics of abundant solutions to the (3+1)-dimensional generalized Yu–Toda–Sasa–Fukuyama equation. Appl. Phys. Lett. 105, 106301 (2020)
Guo, H.D., Xia, T.C., Hu, B.B.: High-order lumps, high-order breathers and hybrid solutions for an extended (3+1)-dimensional Jimbo–Miwa equation in fluid dynamics. Nonlinear Dyn. 100, 601–614 (2020)
Guo, H.D., Xia, T.C., Ma, W.X.: Localized waves and interaction solutions to an extended (3+1)- dimensional Kadomtsev–Petviashvili equation. Mod. Phys. Lett. B. 34, 2050076 (2020)
Kumar, S., Niwasby, M., Wazwa, A.M.: Lie symmetry analysis, exact analytical solutions and dynamics of solitons for (2+1)-dimensional NNV equations. Phys. Scr. (2020). https://doi.org/10.1088/1402-4896/aba5ae
Kumar, S., Kumar, A.: Lie symmetry reductions and group invariant solutions of (2+1)-dimensional modified Veronese web equation. Nonlinear Dyn. 98, 1891–1903 (2019)
Kumar, S., Wazwa, A.M., Kumar, D., Kumar, A.: Group invariant solutions of (2+1)-dimensional rdDym equation using optimal system of Lie subalgebra. Phys. Scr. 94, 115202 (2019)
Kumar, S., Kumar, A., Wazwa, A.M.: New exact solitary wave solutions of the strain wave equation in microstructured solids via the generalized exponential rational function method. Eur. Phys. J. Plus 135, 870 (2020)
Ma, W.X.: Riemann-Hilbert problems of a six-component mKdV system and its soliton solutions. Act. Math. Sci. 39, 509–523 (2019)
Ma, W.X.: Application of the Riemann–Hilbert approach to the multicomponent AKNS integrable hierarchies. Nonlinear Anal. 47, 1–17 (2019)
Ma, W.X.: Riemann–Hilbert problems and N-soliton solutions for a coupled mKdV system. J. Geom. Phys. 132, 45–54 (2018)
Geng, X.G., Wu, J.P.: Riemann-Hilbert approach and N-soliton solutions for a generalized Sasa–Satsuma equation. Wave Motion 60, 62–72 (2016)
Wu, J.P., Geng, X.G.: Inverse scattering transform and soliton classification of the coupled modified Korteweg–de Vries equation. Commun. Nonlin. Sci. Numer. Simulat. 53, 83–93 (2017)
Tian, S.F., Zhang, T.T.: Long-time asymptotic behavior for the Gerdjikov–Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition. Proc. Am. Math. Soc. 146, 1713–1729 (2018)
Wang, D.S., Zhang, D.J., Yang, J.K.: Integrable properties of the general coupled nonlinear Schrödinger equations. J. Math. Phys. 51, 023510 (2010)
Deift, P., Zhou, X.: A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation. Ann. Math. 137, 295–368 (1993)
Fokas, A.S., Lenells, J.: The unified method: I. Nonlinearizable problems on the half-line. J. Phys. A Math. Theor. 45, 195201 (2012)
Lenells, J., Fokas, A.S.: The unified method: II. NLS on the half-line t-periodic boundary conditions. J. Phys. A Math. Theor. 45, 195202 (2012)
Xu, J., Fan, E.G.: The unified transform method for the Sasa–Satsuma equation on the half-line. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469, 20130068 (2013)
Xu, J., Fan, E.G.: Long-time asymptotics for the Fokas-Lenells equation with decaying initial value problem: without solitons. J. Differ. Equ. 259, 1098–1148 (2015)
Wang, D.S., Guo, B.L., Wang, X.L.: Long-time asymptotics of the focusing Kundu–Eckhaus equation with nonzero boundary conditions. J. Differ. Equ. 266, 5209–5253 (2019)
Tian, S.F.: Initial-boundary value problems for the general coupled nonlinear Schrödinger equation on the interval via the Fokas method. J. Differ. Equ. 262, 506–558 (2017)
Yan, Z.Y.: An initial-boundary value problem for the integrable spin-1 Gross–Pitaevskii equations with a \(4\times 4\) Lax pair on the half-line. Chaos 27, 053117 (2017)
Hu, B.B., Xia, T.C.: A Fokas approach to the coupled modified nonlinear Schrödinger equation on the half-line. Math. Methods Appl. Sci. 41, 5112–5123 (2018)
Hu, B.B., Xia, T.C., Ma, W.X.: Riemann-Hilbert approach for an initialboundary value problem of the two-component modified Korteweg–de Vries equation on the half-line. Appl. Math. Comput. 332, 148–159 (2018)
Kumar, D., Kumar, S.: Solitary wave solutions of pZK equation using Lie point symmetries. Eur. Phys. J. Plus 135, 162 (2020)
Kumar, S., Kumarby, A., Kharbanda, H.: Lie symmetry analysis and generalized invariant solutions of (2+1)-dimensional dispersive long wave (DLW) equations. Phys. Scr. 95, 065207 (2020)
Kumar, S., Kumar, M., Kumar, D.: Computational soliton solutions to (2+1)-dimensional Pavlov equation using Lie symmetry approach. Pramana-J. Phys. 94, 28 (2020)
Guo, B.L., Ling, L.M.: Riemann–Hilbert approach and \(N\)-soliton formula for coupled derivative Schrödinger equation. J. Math. Phys. 53, 073506 (2012)
Xiao, Y., Fan, E.G.: A Riemann–Hilbert approach to the Harry–Dym equation on the line. Chin. Ann. Math. Ser. B 37, 373–384 (2016)
Ma, W.X.: Riemann–Hilbert problems of a six-component fourth-order AKNS system and its soliton solutions. Comput. Appl. Math. 37, 6359–6375 (2018)
Kang, Z.Z., Xia, T.C.: Construction of multi-soliton solutions of the \(N\)-coupled Hirota equations in an optical fiber. Chin. Phys. Lett. 36, 110201 (2019)
We would like to express our sincere thanks to editor and reviewers for their valuable comments on this paper. The work was supported in part by the National Natural Science Foundation of China under Grant No. 11975145.
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Guo, HD., Xia, TC. Multi-soliton solutions for a higher-order coupled nonlinear Schrödinger system in an optical fiber via Riemann–Hilbert approach. Nonlinear Dyn 103, 1805–1816 (2021). https://doi.org/10.1007/s11071-020-06166-5
- Riemann–Hilbert approach
- Spectral analysis
- Higher-order coupled nonlinear Schrödinger system
- Soliton solutions