Nonlinear Dynamics

, Volume 95, Issue 3, pp 1957–1964 | Cite as

Excitation control for three-dimensional Peregrine solution and combined breather of a partially nonlocal variable-coefficient nonlinear Schrödinger equation

  • Yi-Xiang ChenEmail author
  • Fang-Qian Xu
  • Yi-Liang Hu
Original Paper


A (2 + 1)-dimensional variable-coefficient partially nonlocal nonlinear Schrödinger equation is considered, and analytical Peregrine solution (PS) and combined Akhmediev breather (AB) are presented from a reduced transformation and the Darboux transformation method. Based on these analytical solutions, the excitation control for three-dimensional PS and combined AB is investigated by comparing values between the maximum value of the effective propagation distance \(\zeta _{\max }\) and the crest position \(\zeta _{0}\) (for the first-order PS and second-order PS) or the crest position \(\zeta _n(n=1,2,3)\) (for the combined AB). If \(\zeta _{\max }<\zeta _{0}\), \(\zeta _{\max }=\zeta _{0}\) and \(\zeta _{\max }>\zeta _{0}\), the anterior excitation, crest excitation, and tail excitation of the first-order PS and second-order PS can be studied, respectively. Similarly, if \(\zeta _{\max }<\zeta _{n}\), \(\zeta _{\max }=\zeta _{n}\) and \(\zeta _{\max }>\zeta _{n}\), the anterior excitation, crest excitation, and tail excitation of the first first-order PS, the second-order PS and the second first-order PS in the combined AB can be discussed, respectively. The \(k+1\)-layer excited structures for the first-order PS, second-order PS, and the combined AB can be constructed for the fixed value of the Hermite polynomial parameter k.


(2 + 1)-Dimensional variable-coefficient nonlinear Partially nonlocal nonlinearity Excitation control Peregrine solution Combined breather 



This work was supported by the National Natural Science Foundation of China (Grant No. 11775185) and the higher school visiting scholar development project (Grant No. FX2013103).

Compliance with ethical standards

Conflict of interest

The authors have declared that no conflict of interest exists.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Electronics InformationZhejiang University of Media and CommunicationsHangzhouPeople’s Republic of China

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