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Rogue wave and multi-pole solutions for the focusing Kundu–Eckhaus Equation with nonzero background via Riemann–Hilbert problem method

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Abstract

In this work, we use the inverse scattering transform method to consider the focusing Kundu–Eckhaus (KE) equation with nonzero background (NZBG) at infinity. Based on the analytical, symmetric, asymptotic properties of eigenfunctions, the inverse problem is solved via a matrix Riemann–Hilbert problem (RHP). The general multi-pole solutions are given in terms of the solution of the associated RHP. And the formula of the N simple-pole soliton solutions are obtained, too. We show that the Peregrine’s rational solution can be viewed as some appropriate limit of the simple-pole soliton solutions at branch point. Furthermore, by taking some other proper limits, the two and three simple-pole soliton solutions can yield to the double- and triple-pole solutions for focusing KE equation with NZBG. The effect of the parameter \(\beta \), characterizing the strength of the non-Kerr (quintic) nonlinear and the self-frequency shift effect, and some typical collisions between solutions of focusing KE equation are graphically displayed.

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Acknowledgements

This work was partially supported by Natural Science Foundation of China under Grant Nos. 11671065, 11971313, 51879045 and Shanghai natural science foundation under Project No. 19ZR1434500.

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  1. College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, People’s Republic of China

    Ning Guo & Jian Xu

  2. School of Mathematical Sciences, Fudan University, Shanghai, 200433, People’s Republic of China

    Lili Wen & Engui Fan

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  1. Ning Guo
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Guo, N., Xu, J., Wen, L. et al. Rogue wave and multi-pole solutions for the focusing Kundu–Eckhaus Equation with nonzero background via Riemann–Hilbert problem method. Nonlinear Dyn 103, 1851–1868 (2021). https://doi.org/10.1007/s11071-021-06205-9

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