Bifurcations and parametric representations of traveling wave solutions for the Green-Naghdi equations (Q2873174)

The paper deals with the traveling wave solutions of the Green-Naghdi equations \(\eta_t+(u\eta)_x=0\), \(u_t+uu_x+\eta_x=\frac{1}{3\eta}\left( \eta^2\frac{d}{dt}(\eta u_x)\right)_x\) arising in the theory of long nonlinear free-surface and interfacial-surface waves. Searching the solutions in the form of the traveling waves \(\eta=\phi(x-ct)\), \(u=\psi(x-ct)\), the equations reduce to a certain system of regular ODEs. The authors present the phase portraits of this system in different parameter regions, study the dynamical behavior of orbits of the associated vector fields, and provide explicit parametric representations of certain traveling wave solutions, particularly, of the solitary wave type.