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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• 作者：Hui-Min Li ; Bo Tian ; Xi-Yang Xie ; Jun Chai ; Lei Liu ; Yi-Tian Gao
• 关键词：(2 + 1) ; dimensional fourth ; order nonlinear Schrödinger equation ; Heisenberg ferromagnetic spin chain ; Generalized Darboux transformation ; Soliton solutions ; Rogue ; wave solutions
• 刊名：Nonlinear Dynamics
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：86
• 期：1
• 页码：369-380
• 全文大小：6,554 KB
• 刊物类别：Engineering
• 刊物主题：Vibration, Dynamical Systems and Control
  Mechanics
  Mechanical Engineering
  Automotive and Aerospace Engineering and Traffic
  
• 出版者：Springer Netherlands
• ISSN：1573-269X
• 卷排序：86

文摘

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 \(\sigma \).