On the solvability of the N-wave, Davey-Stewartson and Kadomtsev-Petviashvili equations

The authors review a rigorous methodology for studying the initial-value problem, for decaying initial data on the plane, for integrable evolution equations in two spatial variables. The N-wave interaction, the Davey-Stewartson and the Kadomtsev-Petviashvili equations are used as illustrative examples. They discuss both the use of a nonlocal Riemann-Hilbert problem and of a delta (DBAR) problem. Some of the results reviewed here are valid only if the initial data satisfy a certain small-norm condition, while some are valid without any small-norm condition.