Fourier-Galerkin method for interacting localized waves

We develop a Fourier-Galerkin spectral technique for computing the solutions of Fourth-order Generalized Wave Equations of type of interacting localized waves. To this end a special complete orthonormal system of functions in L2 ( -oo, oo) is used and a time-stepping algorithm implementing the spectral method is developed. The rate of convergence is shown to be exponential. As a featuring example the head-on collision of sech solitary waves is investigated in the case of Proper Boussinesq Equation (PBE). It is shown that the solitons recover their exact shapes after the collision but experience phase shift. The numerically obtained signs and magnitudes of the phase shifts are in very good quantitative agreement with analytical results for the two soliton solution of PBE.