Numerical investigation of the Klein-Gordon and sine-Gordon equations using the Christov expansion

Abstract The numerical interactions of solitary waves given by the Klein-Gordon and sine-Gordon equations are demonstrated. To do that we apply the “Christov expansion method”, which is a Galerking type expansion. The Christov expansion is based on the Christov functions which are functions that form a complete orthonormal set of functions on L 2 (−∞, ∞) that allow us to expand derivatives, nonlinear products back into the same basis. Here we show how can implement this method to solitary wave equations that form kink or anti-kink type of solutions. We also show how can speed up the algorithms by rewriting the expressions for the nonlinear terms more efficiently. To this end we need to mention that the numerical integration is performed using a Crank–Nicolson type scheme. Our results are in a very good agreement with other published works.