Nonlinear Dynamics

, Volume 90, Issue 3, pp 2221–2230 | Cite as

Nonautonomous multi-peak solitons and modulation instability for a variable-coefficient nonlinear Schrödinger equation with higher-order effects

  • Liu-Ying Cai
  • Xin Wang
  • Lei Wang
  • Min Li
  • Yong Liu
  • Yu-Ying Shi
Original Paper


In this paper, we study a variable-coefficient nonlinear Schrödinger (vc-NLS) equation with fourth-order effects describing an inhomogeneous one-dimensional continuum anisotropic Heisenberg ferromagnetic spin chain or alpha helical protein. The first-order nonautonomous breather solution of the fourth-order vc-NLS equation is derived. The state transition between nonautonomous breather and nonautonomous multi-peak soliton can be realized when group velocity dispersion (GVD) coefficient is proportional to the fourth-order dispersion (FOD) coefficient. We also display how the higher-order effects influence the nonautonomous multi-peak solitons. Our results show that the velocity and localization of the nonautonomous multi-peak soliton are affected by the FOD coefficient, and the peak number is controlled by the GVD coefficient. Further, we also show the compression effect and motion with variable velocity of nonautonomous multi-peak soliton in two kinds of dispersion management systems. Finally, we reveal the relation between the state transition and the modulation instability (MI) analysis.


Variable-coefficient nonlinear Schrödinger equation Nonautonomous multi-peak soliton Higher-order effects Modulation instability 



We express our sincere thanks to all the members of our discussion group for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant (Nos. 11305060, 11271126, 11705290 and 61505054), by the Fundamental Research Funds of the Central Universities (Project No. 2015ZD16), by China Postdoctoral Science Foundation funded sixtieth batches (No. 2016M602252). Lei Wang put forward the idea of this paper. Lei Wang and Liu-ying Cai contributed all mathematical calculation and physical analysis. Lei Wang and Liu-ying Cai wrote the paper. Xin Wang, Min Li, Yong Liu and Yu-ying Shi polished the language. Liu-ying Cai generated all the figures and was responsible for all simulations.


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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of Mathematics and PhysicsNorth China Electric Power UniversityBeijingPeople’s Republic of China
  2. 2.College of ScienceZhongyuan University of TechnologyZhengzhouPeople’s Republic of China
  3. 3.School of Mathematics and StatisticsZhengzhou UniversityZhengzhouPeople’s Republic of China

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