Solitons and periodic waves for the (2 + 1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics

  • Gao-Fu Deng
  • Yi-Tian GaoEmail author
  • Jing-Jing Su
  • Cui-Cui Ding
  • Ting-Ting Jia
Original paper


Fluid mechanics has the applications in a wide range of disciplines, such as oceanography, astrophysics, meteorology, and biomedical engineering. Under investigation in this paper is the (\(2+1\))-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics. Via the Pfaffian technique and certain constraint on the real constant \(\alpha \), the Nth-order Pfaffian solutions are derived. One- and two-soliton solutions are obtained via the Nth-order Pfaffian solutions. Based on the Hirota–Riemann method, one- and two-periodic wave solutions are constructed. With the help of the analytic and graphic analysis, we notice that: (1) of the one soliton, amplitude is irrelevant to \(\gamma \), a real constant coefficient in the equation, velocity along the x direction is independent of \(\gamma \), while velocity along the y direction is proportional to \(\gamma \); (2) one soliton keeps its amplitude and velocity invariant during the propagation and total amplitude of the two solitons in the interaction region is lower than that of any soliton; (3) one-periodic wave can be viewed as a superposition of the overlapping solitary waves, placed one period apart; (4) periodic behaviors for the two-periodic wave exist along the x and y directions, respectively; (5) under certain limiting conditions, one-periodic wave solutions approach to the one-soliton solutions and two-periodic wave solutions approach to the two-soliton solutions.


Fluid mechanics (\(2+1\))-Dimensional generalized Caudrey-Dodd-Gibbon-Kotera-Sawada equation Solitons Periodic waves Pfaffian technique Hirota–Riemann method 



The authors express their sincere thanks to the members of their discussion group for their valuable suggestions. This work has been supported by the National Natural Science Foundation of China under Grant No. 11772017, and by the Fundamental Research Funds for the Central Universities under Grant No. 50100002016105010.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  • Gao-Fu Deng
    • 1
  • Yi-Tian Gao
    • 1
    Email author
  • Jing-Jing Su
    • 1
  • Cui-Cui Ding
    • 1
  • Ting-Ting Jia
    • 1
  1. 1.Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid DynamicsBeijing University of Aeronautics and AstronauticsBeijingChina

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