Nonlinear Dynamics

, Volume 94, Issue 4, pp 2841–2862 | Cite as

Hybrid solutions to Mel’nikov system

  • Xiaoen Zhang
  • Tao Xu
  • Yong Chen
Original Paper


Based on the KP hierarchy reduction technique, explicit two kinds of breather solutions to Mel’nikov system are constructed, one breather is localized in the x-direction and period in the y-direction, the other is the opposite, that is localized in the y-direction and period in the x-direction. Moreover, these two kinds of breather solutions are reduced to the homoclinic orbits and dark soliton or anti-dark soliton solution under suitable parameters constraint respectively. It is interesting that the interaction between the dark soliton and anti-dark soliton is similar to a resonance soliton. In addition, with the long-wave limit, some rational solutions are derived, which possess two different behaviors: lump solution and line rogue wave. Then the dynamics properties of interactions among the obtained solutions are shown through some figures, especially, we not only get the parallel breather but also the intersectional breather during the discussion of the interaction to the two-breather solution. Furthermore, a new three-state interaction composed of dark soliton, rogue wave and breather is generated, this novel pattern is a fantastic phenomenon for the Mel’nikov system.


Mel’nikov system KP hierarchy reduction technique Breather Rational solutions 



Funding was provided by National Natural Science Foundation of China (Grant Nos. 11435005, 11675054).

Compliance with ethical standards

Conflict of interest

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


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Shanghai Key Laboratory of Trustworthy ComputingEast China Normal UniversityShanghaiChina

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