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Periodic Travelling Waves of the Modified KdV Equation and Rogue Waves on the Periodic Background

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Abstract

We address the most general periodic travelling wave of the modified Korteweg–de Vries (mKdV) equation written as a rational function of Jacobian elliptic functions. By applying an algebraic method which relates the periodic travelling waves and the squared periodic eigenfunctions of the Lax operators, we characterize explicitly the location of eigenvalues in the periodic spectral problem away from the imaginary axis. We show that Darboux transformations with the periodic eigenfunctions remain in the class of the same periodic travelling waves of the mKdV equation. In a general setting, there exist three symmetric pairs of simple eigenvalues away from the imaginary axis, and we give a new representation of the second non-periodic solution to the Lax equations for the same eigenvalues. We show that Darboux transformations with the non-periodic solutions to the Lax equations produce rogue waves on the periodic background, which are either brought from infinity by propagating algebraic solitons or formed in a finite region of the time-space plane.

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Acknowledgements

The work of J.C. was supported by the National Natural Science Foundation of China (No.11471072). The work of D.P. is supported by the State task program in the sphere of scientific activity of Ministry of Education and Science of the Russian Federation (Task No. 5.5176.2017/8.9) and from the grant of President of Russian Federation for the leading scientific schools (NSH-2685.2018.5).

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Authors and Affiliations

  1. School of Mathematics, Southeast University, Nanjing, 210096, Jiangsu, People’s Republic of China

    Jinbing Chen

  2. Department of Mathematics, McMaster University, Hamilton, ON, L8S 4K1, Canada

    Dmitry E. Pelinovsky

  3. Department of Applied Mathematics, Nizhny Novgorod State Technical University, 24 Minin street, Nizhny, Novgorod, Russia, 603950

    Dmitry E. Pelinovsky

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  1. Jinbing Chen
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  2. Dmitry E. Pelinovsky
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Correspondence to Dmitry E. Pelinovsky.

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Communicated by Dr. Paul Newton.

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Chen, J., Pelinovsky, D.E. Periodic Travelling Waves of the Modified KdV Equation and Rogue Waves on the Periodic Background. J Nonlinear Sci 29, 2797–2843 (2019). https://doi.org/10.1007/s00332-019-09559-y

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  • DOI: https://doi.org/10.1007/s00332-019-09559-y

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