Symplectic computation of solitary waves for general Sine–Gordon equations

We present a class of symplectic schemes for the computation of solutions of the general Sine–Gordon systems. These difference methods are constructed based on the symplectic schemes to the infinite-dimensional Hamiltonian system via generating functions. We use these schemes to compute several types of waves phenomena for the Sine–Gordon and the Φ4 systems. It is demonstrated that these schemes preserve the long-time global structure and the topological stabilities of solitary waves very well. Also, our numerical results indicate that these schemes can be used as effective tools for the numerical investigations of the solutions of general Sine–Gordon equations.