We deal with numerical analysis and simulations of the Davey-Stewartson equations
which model, for example, the evolution of water surface waves.
This time dependent PDE system is particularly interesting as a generalization
of the 1-d integrable NLS to 2 space dimensions.
We use a time splitting spectral method where
we give a convergence analysis for the semi-discrete version of the scheme.
Numerical results are presented for various blow-up phenomena of
the equation, including blowup of defocusing,
elliptic-elliptic Davey-Stewartson systems
and simultaneous blowup at multiple locations in the focusing
Also the modeling of exact soliton type solutions
for the hyperbolic-elliptic (DS2) system is studied.