Numerical solutions of nonlinear wave equations

Accurate, stable numerical solutions of the (nonlinear) sine-Gordon equatio n are obtained with particular consideration of initial conditions that are exponentially close to the phase space homoclinic manifolds. Earlier local , grid-based numerical studies have encountered difficulties, including num erically induced chaos for such initial conditions. The present results are obtained using the recently reported distributed approximating functional method for calculating spatial derivatives to high accuracy and a simple, e xplicit method for the time evolution. The numerical solutions are chaos-fr ee for the same conditions employed in previous work that encountered chaos . Moreover, stable results that are free of homoclinic-orbit crossing are o btained even when initial conditions are within 10(-7) of the phase space s eparatrix value pi. It also is found that the present approach yields extre mely accurate solutions for the Korteweg-de Vries and nonlinear Schrodinger equations. Our results support Ablowitz and co-workers' conjecture that en suring high accuracy of spatial derivatives is more important than the use of symplectic time integration schemes for solving solitary wave equations. [S1063-651X(99)02601-X].