Nonlinear Dynamics

, Volume 94, Issue 4, pp 2563–2573 | Cite as

High-order revivable complex-valued hyperbolic-sine-Gaussian solitons and breathers in nonlinear media with a spatial nonlocality

  • Zhen-Jun YangEmail author
  • Shu-Min ZhangEmail author
  • Xing-Liang Li
  • Zhao-Guang Pang
  • Hong-Xia Bu
Original Paper


In this paper, based on the nonlocal nonlinear Schrödinger equation, the evolution of complex-valued hyperbolic-sine-Gaussian beams (CVHSGBs) is investigated in nonlinear media with a spatial nonlocality. It is found that the evolution of CVHSGBs is variable depending on the parameters of complex-valued hyperbolic sine function. Choosing special parameters, the pattern of CVHSGBs can keep unchanged during propagation, and they propagate as solitons or breathers. Furthermore, for the general case, the CVHSGB evolutes periodically, and it recovers into its initial pattern at the end of each evolution period, namely it can be revivable periodically, which can be regarded as a generalized high-order breather. A series of analytical expressions are derived to describe the beam evolution, the intensity pattern, the beam spot size, the real beam curvature, etc. Some numerical simulations are also performed to demonstrate the typical evolution properties.


Nonlocal nonlinear Schrödinger equation Nonlocal media Nonlinear propagation Soliton Breather 



This research was supported by the National Natural Science Foundation of China (Grant Nos. 61308016, 11374089, and 61605040), the Chunhui Plan of Ministry of Education of China (Grant No. Z2017020), the Natural Science Foundation of Hebei Province (Grant Nos. F2017205060, F2017205162, and F2016205124), the Technology Key Project of Colleges and Universities of Hebei Province (Grant Nos. ZD2018081 and ZD2016031), and the Science Fund for Distinguished Young Scholars of Hebei Normal University (Grant No. L2017J02).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest concerning the publication of this manuscript.


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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.College of Physics and Information Engineering, Hebei Advanced Thin Films Key LaboratoryHebei Normal UniversityShijiazhuangChina

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