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Riemann–Hilbert approach and nonlinear dynamics of the coupled higher-order nonlinear Schrödinger equation in the birefringent or two-mode fiber

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Abstract

The multi-soliton solutions and breathers to the coupled higher-order nonlinear Schrödinger (CH-NLS) equation are derived in this work via the Riemann–Hilbert approach. Firstly, the spectral structure of the CH-NLS equation is investigated and then a matrix Riemann–Hilbert problem on the real axis is strictly formulated. Secondly, by solving the special Riemann–Hilbert problem with no reflection where a jump matrix is taken to be the identity matrix, the formula of N-soliton solutions can be computed. Thirdly, we prove that the higher-order linear and nonlinear term r has important impact on the velocity, phase, period and wavewidth of wave dynamics. Besides, the localized waves characteristics together with collision dynamic behaviors of these explicit soliton solutions and breathers are shown graphically and discussed in detail. Interestingly, three solitons display different dynamics which demonstrate amplitudes of the right-direction waves gradually become larger during the propagation process.

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Acknowledgements

The authors thank Yan Zhang for helpful advices during the writing of this paper. The authors also would like to express their sincere thanks to the anonymous editor and reviewers for their valuable comments on this paper. The work is supported by the National Natural Science Foundation of China (No. 12026245, 11671095), the Natural Science Foundation of Henan (No. 202300410524) and the Science and Technique Project of Henan (No. 212102310397).

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Authors and Affiliations

  1. College of Mathematics and Statistics, Zhoukou Normal University, Zhoukou, 466001, China

    Han-Yu Wei

  2. School of Mathematical Science, Fudan University, Shanghai, 200433, China

    Han-Yu Wei & En-Gui Fan

  3. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Han-Dong Guo

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  1. Han-Yu Wei
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  2. En-Gui Fan
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  3. Han-Dong Guo
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Correspondence to Han-Yu Wei.

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Wei, HY., Fan, EG. & Guo, HD. Riemann–Hilbert approach and nonlinear dynamics of the coupled higher-order nonlinear Schrödinger equation in the birefringent or two-mode fiber. Nonlinear Dyn 104, 649–660 (2021). https://doi.org/10.1007/s11071-021-06286-6

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