Inverse Scattering in Two Dimensions

Grants and Contracts Details


The principal investigator is working in two-dimensional inverse scattering for Schrodinger and Dirac-type systems associated to completely integrable, dispersive nonlirn3ar partial differential equations in two space and one time dimensions. Recent developments in multilinear harmonic analysis-including Brascamp-Lieb type inequalities-make these completely integrable systems more amenable to rigorous analysis. The inverse scattering method consists of three parts: (i) a scattering transform which linearizes the nonlinear evolution defined by the PDE, (2) a linear evolution equation for the scattering transform, and (3) an inverse scattering transform which produces the solution to the PDE from the time-evolved scattering data. For completely integrable evolution equations in one space dimension (such as the KdV or cubic NLS equations), the direct scattering transform is defined by the scattering data associated to a linear Schrodinger or Dirac-type equation where the solution o1f the nonlinear PDE enters as a coefficient. The inverse scattering map is defined by a Riemann-Hilbert problem in which the space and time variables enter as parameters in a phase function. The asymptotic analysis of solutions has been raised to a high art thanks to the "nonlinear stationary phase" method of Deift and Zhou. For the completely integrable evolution equations in two space dimensions to be studied here, the inverse scattering formalism has a similar structure, but the inverse map defined by a d-bar problem or a non-local Riemann-Hilbert problem. These problems are ifar less well-understood than the Riemann-Hilbert problems of one-dimensional inverse theory. The proposed research will develop tools for asymptotic analysis of these d-bar and non-local Riemann-Hilbert problems by focussing on carefully chosen model problems. Together with collaborators and doctoral students, the Pl has studied the scattiering maps associated to the Davey-Stewartson II (OS II) and Novikov-Veselov (NV) equations. Rigorous analysis of the scattering maps--as defined by linear spectral problems--has already led to global existence (defocussing DS II, NV) and large-time asymptotics (OS II). In neither case were global 'existence results previously known, nor do they appear, at least for the present, to be accessible using PDE techniques. The Pl is also working with collaborators in the Finnish Centre for Excellence in Inverse Problems in Helsinki to study d-bar and non-local Riemann Hilbert problems wiith a similar structure with applications to medical and seismic imaging.
Effective start/end date9/1/158/31/22


  • Simons Foundation


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