By using the bifurcation theory of dynamical systems to study the dynamical behavior of the Green–Naghdi equations, the existence of solitary wave solutions along with smooth periodic traveling wave solutions is obtained. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact and explicit parametric representations of traveling wave solutions are constructed. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 970 – 980 ¨ ITAK ˙ c TUB ⃝ doi: Bifurcations and parametric representations of traveling wave solutions for the Green–Naghdi equations Shengqiang TANG,∗ Libing ZENG, Dahe FENG School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, . China Received: • Accepted: • Published Online: • Printed: Abstract: By using the bifurcation theory of dynamical systems to study the dynamical behavior of the Green–Naghdi equations, the existence of solitary wave solutions along with smooth periodic traveling wave solutions is obtained. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact and explicit parametric representations of traveling wave solutions are constructed. Key words: Green–Naghdi equations, bifurcation theory of dynamical systems, bifurcation curves, solitary waves, periodic waves 1. Introduction In this paper, we study the dynamical behavior of the Green–Naghdi (GN) equations [3, 4]. Specifically, we determine traveling wave solutions and new solitary wave solutions for the GN equations: ηt + (uη)x = 0, ut + uux + ηx = 1 3η ( ) d η 2 (ηux ) . dt x () () The Green–Naghdi equations were derived for both free-surface and interfacial-surface waves under the assumption of long wavelengths. Here, η and u represent the surface disturbance and the mean horizontal velocity, respectively. The GN equations were originally developed by Green and Naghdi in 1974 to analyze some nonlinear free-surface flows. After the successful application of the GN equations to nonlinear ship wave-making problems [2], the method was applied to many nonlinear water wave problems. Later, the model was extended to deep-water waves by Webster and Kim and by
