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

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Abstract

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.

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Acknowledgements

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

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Authors and Affiliations

  1. School of Computer Information Management, Inner Mongolia University of Finance and Economics, Hohhot, 010070, China

    Zhong-Zhou Lan

  2. Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China

    Jing-Jing Su

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This work has been supported by the Science Research Project of Higher Education in Inner Mongolia Autonomous Region under Grant No. NJZZ18117, by the Natural Science Foundation of Inner Mongolia Autonomous Region under Grant No. 2018BS01004, by the China Postdoctoral Science Foundation under Grant No. 2018M640094, by the Program for Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region under Grant No. NJYT-19-B21, and by the National Natural Science Foundation of China under Grant No. 11772017.

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For Mathematica code download please log in: https://github.com/ouyangyalin0312/Solitary-and-rogue-waves-solutions-for-NA-GAB-system.

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Lan, ZZ., Su, JJ. Solitary and rogue waves with controllable backgrounds for the non-autonomous generalized AB system. Nonlinear Dyn 96, 2535–2546 (2019). https://doi.org/10.1007/s11071-019-04939-1

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