Nonlinear Dynamics

, Volume 98, Issue 3, pp 1731–1744 | Cite as

Higher-order interactional solutions and rogue wave pairs for the coupled Lakshmanan–Porsezian–Daniel equations

  • Tao XuEmail author
  • Guoliang He
Original Paper


The N-fold Darboux transformation of the coupled Lakshmanan–Porsezian–Daniel (LPD) equations is constructed. Based on the Darboux transformation and the limiting technique, we investigate two kinds of solutions for the coupled LPD equations, which are higher-order interactional solutions and rogue wave (RW) pairs. Through considering the double-root situation of the spectral characteristic equation for the matrix in the Lax pair, we give the higher-order interactional solutions among higher-order RWs, multi-bright (dark) solitons and multi-breather. Besides, we consider the triple-root situation of the spectral characteristic equation and get the higher-order RW pairs. It demonstrates that the RW pairs are greatly different form the traditional higher-order RWs. The fist-order RW pairs can split into two traditional first-order RWs, and four or six traditional fundamental RWs can emerge from the second-order case. The corresponding dynamics of these explicit solutions are discussed in detail.


The coupled Lakshmanan–Porsezian–Daniel equations Darboux transformation Interactional solutions Rogue waves Breathers Solitons Rogue wave pairs 



This work is supported by National Natural Science Foundation of China (Grant No. 11871232) and the Training Plan of Young Key Teachers in Universities of Henan Province.

Compliance with ethical standards

Conflict of interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceZhengzhou University of Light IndustryZhengzhouPeople’s Republic of China

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