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Solitons and rogue waves of the quartic nonlinear Schrödinger equation by Riemann–Hilbert approach

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Abstract

We systematically develop a Riemann–Hilbert approach for the quartic nonlinear Schrödinger equation on the line with both zero boundary condition and nonzero boundary conditions at infinity. For zero boundary condition, the associated Riemann–Hilbert problem is related to two cases of scattering data: N simple poles and one Nth-order pole, which allows us to find the exact formulae of soliton solutions. In the case of nonzero boundary conditions and initial data that allow for the presence of discrete spectrum, the pure one-soliton solution and rogue waves are presented. The important advantage of this method is that one can study the long-time asymptotic behavior of the solutions and the infinite order rogue waves based on the associated Riemann–Hilbert problems.

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Acknowledgements

N. Liu is supported by the China Postdoctoral Science Foundation under Grant Nos. 2019TQ0041 and 2019M660553.

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  1. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, People’s Republic of China

    Nan Liu & Boling Guo

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  1. Nan Liu
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Correspondence to Nan Liu.

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Liu, N., Guo, B. Solitons and rogue waves of the quartic nonlinear Schrödinger equation by Riemann–Hilbert approach. Nonlinear Dyn 100, 629–646 (2020). https://doi.org/10.1007/s11071-020-05521-w

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