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Abstract
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.
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