Numerical solutions of nonlinear wave equations

Abstract

Accurate, stable numerical solutions of the (nonlinear) sine-Gordon equation 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 numerically 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, explicit method for the time evolution. The numerical solutions are chaos-free for the same conditions employed in previous work that encountered chaos. Moreover, stable results that are free of homoclinic-orbit crossing are obtained even when initial conditions are within ${10}^{\ensuremath{-}7}$ of the phase space separatrix value \ensuremath{\pi}. It also is found that the present approach yields extremely accurate solutions for the Korteweg--de Vries and nonlinear Schr\"odinger equations. Our results support Ablowitz and co-workers' conjecture that ensuring high accuracy of spatial derivatives is more important than the use of symplectic time integration schemes for solving solitary wave equations.