# American Institute of Mathematical Sciences

## Smooth and singular traveling wave solutions for the Serre-Green-Naghdi equations

 1 College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China 2 African Institute for Mathematical Sciences, Muizenberg, Cape Town, South Africa 3 Department of Mathematics, Shanghai Normal University, Shanghai 200234, China 4 Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

* Corresponding author: Yixia Shi

Received  February 2019 Revised  June 2019 Published  December 2019

Fund Project: Part of this work was done during L. Zhang's research visit to African Institute for Mathematical Sciences South Africa

In this paper, we consider the traveling wave solutions of the one-dimensional Serre-Green-Naghdi (SGN) equations which are proposed to model dispersive nonlinear long water waves in a one-layer flow over flat bottom. We decouple the traveling wave system of SGN equations into two ordinary differential equations. By studying the bifurcations and phase portraits of each bifurcation set of one equation, we obtain the exact traveling wave solutions of SGN equations for the variable $u(x, t)$ which represents average horizontal velocity of water wave. For the compacted orbits intersecting with the singular line in phase plane, we obtained two families of solutions: a family of smooth traveling wave solutions including periodic wave solutions and solitary wave solutions, and a family of compacted singular solutions which have continuous first-order derivative but discontinuous second-order derivative.

Citation: Lijun Zhang, Yixia Shi, Maoan Han. Smooth and singular traveling wave solutions for the Serre-Green-Naghdi equations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020217
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##### References:
The phase portraits of system (9) in each bifurcation set for $A>0$. (1) $c>\frac{B}{A}-\frac{3}{2}\sqrt{Ag}$; (2) $c = \frac{B}{A}-\frac{3}{2}\sqrt{Ag}$; (3) $c<\frac{B}{A}-\frac{3}{2}\sqrt{Ag}$
Phase orbit of (12) and the corresponding bounded traveling wave solutions. (1) a compact orbit of (12) intersecting with the singular line; (2) a smooth periodic traveling wave solution; (3) compacton
Phase orbit of (12) and the corresponding bounded traveling wave solutions. (1) a compact orbit of (12) intersecting with the singular line; (2) a smooth periodic traveling wave solution; (3) compacton
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