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Soliton and rogue-wave solutions for a (2 + 1)-dimensional fourth-order nonlinear Schrödinger equation in a Heisenberg ferromagnetic spin chain

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Abstract

In this paper, we investigate the soliton and rogue-wave solutions for a (2 + 1)-dimensional fourth-order nonlinear Schrödinger equation, which describes the spin dynamics of a Heisenberg ferromagnetic spin chain with the bilinear and biquadratic interactions. For such an equation, there exists a gauge transformation which converts the nonzero potential Lax pair into some constant-coefficient differential equations. Solving those equations, vector solutions for the nonzero potential Lax pair are obtained. The condition for the modulation instability of the plane-wave solution is also given through the linear stability analysis. Then, we present the determinant representations for the N-soliton solutions via the Darboux transformation (DT) and Nth-order rogue-wave solutions via the generalized DT. Profiles for the solitons and rogue waves are analyzed with respect to the lattice parameter \(\sigma \), respectively. When \(\sigma \) is greater than a certain value marked as \(\sigma _{0}\), one-soliton velocities increase with the increase of \(\sigma \). When \(\sigma <\sigma _{0}\), one-soliton velocities decrease with the increase of \(\sigma \). When the time t is equal to zero, \(\sigma \) has no effect on the interactions between the two solitons. When \(t\ne 0\), different choices of \(\sigma \) lead to the different two-soliton velocities, giving rise to the different interaction regions. Widths of the first-order rogue waves become bigger with the decrease of \(\sigma \), while the amplitudes do not depend on \(\sigma \). The second-order rogue waves are composed by three first-order rogue waves whose widths all get wider with the decrease of \(\sigma \), while the amplitudes do not depend on \(\sigma \).

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Acknowledgments

This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11272023, 11271362, 11201501, 11375030 and 11571389, Beijing Natural Science Foundation under Grant No. 1153004, Beijing Nova program 45 No. Z131109000413029, Beijing Finance Funds of Natural Science Program for Excellent Talents No. 2014000026833ZK19 and Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications).

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Authors and Affiliations

  1. State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Hui-Min Li, Bo Tian, Xi-Yang Xie, Jun Chai & Lei Liu

  2. Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China

    Yi-Tian Gao

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  1. Hui-Min Li
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  2. Bo Tian
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Correspondence to Bo Tian.

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Li, HM., Tian, B., Xie, XY. et al. Soliton and rogue-wave solutions for a (2 + 1)-dimensional fourth-order nonlinear Schrödinger equation in a Heisenberg ferromagnetic spin chain. Nonlinear Dyn 86, 369–380 (2016). https://doi.org/10.1007/s11071-016-2894-y

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