The d-bar method: Inverse scattering, nonlinear waves, and random matrices (12frg176)

Objectives

By bringing together experts in integrable systems, harmonic analysis,
scattering theory, and nonlinear PDE's, we will build on recent progress in
the theory of the $overline{partial}$-problem first of all to study
well-posedness and space-time asymptotics of solutions to completely
integrable, nonlinear dispersive equations in two-dimensions.

(1) We will begin with the defocussing Davey-Stewartson equation, for which
the inverse scattering map is well-understood and solutions are known to be
well-posed. The inverse scattering solution can be represented as a finite sum
of multilinear integrals with oscillating phase plus a controllable remainder;
building on techniques of Bennett, Carbery, Christ, and Tao cite{BCCT:2010}
and Christ, Li, Tao and Thiele to obtain temporal decay, and attempt to
develop these techniques further into a precise analytical tool for studying
oscillatory $overline{partial}$-problems that arise for related equations
such as the Novikov-Veselov equation. We will also attack model problems
involving soliton behavior.

(2) We will consider initial-boundary problems for the Davey-Stewartson
equation and Kadomtsev-Petviashvili equation on $mathbb{R}_{x}^{+}%
timesmathbb{R}_{t}^{+}$, which are expected to exhibit a richer dynamical
structure. These problems pose the additional challenge of understanding how
all of the boundary data are determined from a subset that makes the problem
by means of the textquotedblleft global relationtextquotedblright (see the
expository paper cite{Fokas:2010} and see, for example, cite{Fokas:2010b},
cite{MF:2011} for recent work respectively on the Davey-Stewartson and
Kadomtsev-Petviashvili equations on $mathbb{R}_{x}^{+}timesmathbb{R}%
_{t}^{+}$).

(3) In a complementary effort, we will use techniques of Kenig, Ponce, and
Vega cite{KPV:1991} to develop the well-posedness theory for third-order
dispersive nonlinear equations such as the Novikov-Veselov equation. The
well-posedness theory will provide a framework for further investigation of
the Novikov-Veselov equation and related equations such as the KP system,
which may be obtained from the NV equation by a limiting procedure.

Although our efforts in this focussed research group activity will primarily
concern dispersive nonlinear PDE's, we will build on our experience in
asymptotic problems for PDE's to attack related problems in normal matrix
theory and orthogonal polynomials in the plane.

References for Overview and Objectives

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