Travelling Wave Solutions to Stretched Beam's Equation: Phase Portraits Survey

In this paper, following the phase portraits ana/ysis, we investigate the integrabih'ty of a system which physically describes the transverse oscillation of an elastic beam under end-thrust. As a result, we find that this system actually comprises two families of travelling waves: the sub- and super-sonic periodic waves of positive- and negative-definite velocities, respectively, and the localized sub-sonic loop-shaped waves of positive-definite velocity. Expressing the energy-like of this system while depicting its phase portrait dynamics, we show that these multivalued localized travelling waves appear as the boundary solutions to which the periodic travelling waves tend asymptotically.