Fourth order time-stepping for Kadomtsev-Petviashvili and Davey-Stewartson equations

Purely dispersive partial differential equations as the Korteweg-de Vries equation, the nonlinear Schr"odinger equation and higher dimensional generalizations thereof can have solutions which develop a zone of rapid modulated oscillations in the region where the corresponding dispersionless equations have shocks or blow-up. To numerically study such phenomena, fourth order time-stepping in combination with spectral methods is beneficial to resolve the steep gradients in the oscillatory region. We compare the performance of several fourth order methods for the Kadomtsev-Petviashvili and the Davey-Stewartson equations, two integrable equations in 2+1 dimensions: exponential time-differencing, integrating factors, time-splitting, implicit Runge-Kutta and Driscoll’s composite Runge-Kutta method. The accuracy in the numerical conservation of integrals of motion is discussed.