  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.
