Fourier-Galerkin spectral method for localized solutions of nonlinear equations.

摘要

We develop a Fourier-Galerkin spectral technique to compute localized solutions of equations with cubic nonlinearity and equations with higher order dispersion terms. Moreover, we compute the solutions of interacting localized waves type of the Fourth-Order Generalized Wave Equations. To this end, a special complete orthonormal system of functions in L 2(-infinity,infinity) is used, and a time-stepping algorithm implementing the spectral method is developed. The rate of convergence is shown to be exponential.; As featuring examples for localized solutions we investigate the Boussinesq Equation with Cubic Nonlinearity and the Sixth-Order Generalized Boussinesq Equation. In the case of interacting localized waves, we are investigating the head-on collision of sech-type solitary waves for the Proper Boussinesq Equation (PBE). We show 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 the PBE.