Kawahara solitons in Boussinesq equations using a robust Christov-Galerkin spectral method

We develop a robust Christov–Galerkin spectral technique for computing interacting localized wave solutions of and fourth and sixth-order generalized wave equations. To this end, a special complete orthonormal system of functions in Lð 1;1Þ is used whose rate of convergence is shown to be exponential for the cases under consideration. For the time-stepping, an implicit algorithm is chosen which makes use of the banded structure of the matrices representing the different spatial derivatives. As featuring examples, the head-on collision of solitary waves is investigated for a sixthorder generalized Boussinesq equation and a fourth-order Boussinesq type equation with a linear term. Its solutions comprise monotone shapes (sech-es) and damped oscillatory shapes (Kawahara solitons). The numerical results are validated against published data in the literature using the method of variational imbedding. 2014 Elsevier Inc. All rights reserved.