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

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7 Citations (Scopus)


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 L2(-∞,∞) 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.

Original languageEnglish
Pages (from-to)245-257
Number of pages13
JournalApplied Mathematics and Computation
Publication statusPublished - 15 Sep 2014


  • Boussinesq equations
  • Kawahara solitons
  • Soliton interaction
  • Spectral methods


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