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

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Abstract

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.

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Acknowledgements

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.

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Authors and Affiliations

  1. Department of Physics, Honghe University, Mengzi, 661100, Yunnan, People’s Republic of China

    Qing Meng

  2. College of Mathematics, Honghe University, Mengzi, 661100, Yunnan, People’s Republic of China

    Bin He & Wei Liu

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  1. Qing Meng
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  2. Bin He
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  3. Wei Liu
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Correspondence to Qing Meng.

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Meng, Q., He, B. & Liu, W. Exact similarity and traveling wave solutions to an integrable evolution equation for surface waves in deep water. Nonlinear Dyn 92, 827–842 (2018). https://doi.org/10.1007/s11071-018-4093-5

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