Nonlinear Dynamics

, Volume 96, Issue 4, pp 2535–2546 | Cite as

Solitary and rogue waves with controllable backgrounds for the non-autonomous generalized AB system

  • Zhong-Zhou LanEmail author
  • Jing-Jing SuEmail author
Original Paper


Investigated in this paper is a non-autonomous generalized AB system, which is used to describe certain baroclinic instability processes in the geophysical flows. We discover that the two short waves and mean flow can evolve in the forms of the multi-rogue waves on the condition that the nonlinearity effect \(\sigma \) is positive. Via the Darboux and generalized Darboux transformations, we obtain the first- and second-order rogue waves as well as the algorithm to derive the Nth-order rogue waves. It is revealed that the perturbation function \(\delta (t)\) has no effect on the two short waves while affects the mean flow by changing its evolution background. When \(\sigma \) is negative, those rogue waves turn to be singular. In addition, we find that the two short waves and mean flow can also appear as the solitary waves, and they perform as the “bright” solitons under \(\sigma >0\) while perform as the “dark” solitons under \(\sigma <0\). With the Hirota method, introducing the auxiliary function \(\alpha (t)\), we derive the first- and second-order bright and dark solitary waves. Both solitary wave velocities are related to \(\delta (t)\) and \(\alpha (t)\). Besides, \(\delta (t)\) and \(\alpha (t)\) have no effect on the amplitudes of the two short waves but bring about controllable backgrounds and deformations of the solitary waves for the mean flow.


Geophysical flows Non-autonomous generalized AB system Solitary and rogue waves Controllable backgrounds 



We express our sincere thanks to all the members of our discussion group for their valuable comments.

Compliance with ethical standards

Conflict of interest

No potential conflict of interest was reported by the authors.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Computer Information ManagementInner Mongolia University of Finance and EconomicsHohhotChina
  2. 2.Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid DynamicsBeijing University of Aeronautics and AstronauticsBeijingChina

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