Coherent solutions for the fundamental resonance of the Boussinesq equation

We consider the Boussinesq equation in an infinite wall under an external resonant weak excitation. Two slow flow equations are obtained and amplitude and phase modulation equations as well as external force-response and frequency–response curves are determined. Energy considerations are used in order to study the global behavior of the two slow flow equations and to demonstrate the existence of closed orbits around the equilibrium points with a second frequency in addition to the forcing frequency. Two period quasi-periodic motions are present with amplitudes depending on the initial conditions. Moreover, in certain cases librations can occur with a frequency depending on the amplitude of the external excitation. If the external excitation increases, the modulation period for the slow flow equations becomes infinite and an infinite-period bifurcation occurs, while the modulation amplitude remains finite and the libration transforms into a closed orbit. Finally, we demonstrate that the separatrix of the two slow flow equations corresponds to an asymptotically steady-state and phase-locked periodic solution.