High-order rogue waves and their dynamics of the Fokas–Lenells equation revisited: a variable separation technique

  • Zihao Wang
  • Linyun He
  • Zhenyun QinEmail author
  • Roger Grimshaw
  • Gui Mu
Original paper


The Fokas–Lenells (FL) equation is an integrable higher-order extension of nonlinear Schrödinger equation. One approach to generating its breather solutions is based on Darboux transformation (DT) and iterations. However, the DT of FL equation contains negative powers of the spectral parameter, which can lead to very complicated expressions when N is large. In this paper, we avoid the negative powers by adopting a variable separation and Taylor expansion technique to solve the Lax pair of FL system. Furthermore, stability of the proposed technique is demonstrated in detail.


Fokas–Lenells equation Lax pairs Variable separation Rogue waves 



This work is sponsored by the National Natural Science Foundation of China (No. 11571079, No. 11801153, No. 11701322), Shanghai Pujiang Program (No. 14PJD007), the Natural Science Foundation of Shanghai (No. 14ZR1403500) and the Young Teachers Foundation (No. 1411018) of Fudan University. Also, the authors are very grateful to Professor Peter D. Miller and Professor John E. Fornaess for their enthusiastic support and useful suggestions.

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Zihao Wang
    • 1
    • 2
  • Linyun He
    • 1
    • 3
  • Zhenyun Qin
    • 4
    Email author
  • Roger Grimshaw
    • 5
  • Gui Mu
    • 6
  1. 1.School of Mathematical SciencesFudan UniversityShanghaiPeople’s Republic of China
  2. 2.Engineering Science and Applied Mathematics DepartmentNorthwestern UniversityEvanstonUSA
  3. 3.Harold and Inge Marcus Department of Industrial and Manufacturing EngineeringPennsylvania State UniversityPennsylvaniaUSA
  4. 4.School of Mathematical Sciences and Key Laboratory of Mathematics for Nonlinear ScienceFudan UniversityShanghaiPeople’s Republic of China
  5. 5.Department of MathematicsUniversity College LondonLondonUK
  6. 6.School of MathematicsKunming UniversityKunmingPeople’s Republic of China

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