Nonlinear Dynamics

, Volume 92, Issue 3, pp 827–842 | Cite as

Exact similarity and traveling wave solutions to an integrable evolution equation for surface waves in deep water

  • Qing Meng
  • Bin He
  • Wei Liu
Original Paper


In this paper, the Lie symmetry analysis and the dynamical system method are performed on an integrable evolution equation for surface waves in deep water
$$\begin{aligned} 2\sqrt{\frac{k}{g}}u_{xxt}=k^2u_x-\frac{3}{2}k(uu_x)_{xx}. \end{aligned}$$
All of the geometric vector fields of the equation are presented, as well as some exact similarity solutions with an arbitrary function of t are obtained by using a special symmetry reduction and the dynamical system method. Different kinds of traveling wave solutions also be found by selecting the function appropriately.


Integrable evolution equation Symmetry reduction Qualitative analysis Similarity solution Traveling wave solution 



This work is supported by the National Natural Science Foundation of China under Grant No. 11461022 and the Natural Science Major Foundation of Yunnan Province, China, under Grant No. 2014FA037.

Compliance with ethical standards

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.

Supplementary material

11071_2018_4093_MOESM1_ESM.rar (2.6 mb)
Supplementary material 1 (rar 2691 KB)


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of PhysicsHonghe UniversityMengziPeople’s Republic of China
  2. 2.College of MathematicsHonghe UniversityMengziPeople’s Republic of China

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