Generalizing the celebrated KdV and NLS equations, the Novikov-Veselov (NV) and Davey-Stewartson (DS) equations are completely integrable, nonlinear, dispersive partial differential equations in two space and one time dimensions. They arise as limits of the water wave equations for two-dimensional surface waves. I will report on recent progress in the rigorous analysis of the direct and inverse scattering maps which, formally, conjugate these equations to a simple linear flow. Using the inverse scattering method, one can then describe the detailed asymptotic behavior of the solutions.
For both the NV and DS equations, the inverse scattering maps are defined by $\overline\partial$-problems, first formulated by Fokas-Ablowitz and Beals-Coifman, in which the space and time variables enter as parameters. Large-parameter asymptotics govern the behavior of the dynamical system. Analogous $\overline\partial $-problems govern asymptotics of orthogonal polynomials in the plane, and random normal matrix distributions. The long-term goal of this research program is to develop the $\overline\partial$-method for two-dimensional integrable systems into a rigorous asymptotic method to approach these problems analogous to the Riemann-Hilbert method for one-dimensional integrable systems.
Part of the work described is joint work with Michael Music and Samuli Siltanen.
