1. Kawahara solitons in Boussinesq equations using a robust Christov–Galerkin spectral method
- Author
-
N. C. Papanicolaou and M. A. Christou
- Subjects
Computational Mathematics ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Monotone polygon ,Rate of convergence ,Applied Mathematics ,Mathematical analysis ,Orthonormal basis ,Boussinesq approximation (water waves) ,Galerkin method ,Wave equation ,Spectral method ,Exponential function ,Mathematics - Abstract
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 L^2(-~,~) 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 sixth-order 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.
- Published
- 2014