Article Contents
Article Contents

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

• * Corresponding author: Yixia Shi
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.

Mathematics Subject Classification: 35C07, 34G20, 34C23.

 Citation:

• Figure 1.  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) $c<\frac{B}{A}-\frac{3}{2}\sqrt{Ag}$

Figure 2.  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

Figure 3.  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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