Scattering and Inverse Scattering in Multidimensions, May 15-23, 2014

With the generous support of the National Science Foundation (NSF) and the Institute for Mathematics and its Applications (IMA),  the Department of Mathematics at the University of Kentucky will host a nine-day conference and workshop on "Scattering and Inverse Scattering in Multidimensions" organized by Kenneth T.-R. McLaughlin (University of Arizona), Peter D. Miller (University of Michigan), and Peter A. Perry (University of Kentucky). The purpose of this conference is to bring together researchers in the related fields of dispersive partial differential equations, completely integrable systems, and inverse problems to make progress on fundamental unsolved problems of two-dimensional, completely integrable systems including integrable dispersive PDE's and normal matrix distributions.

The deadline for registration is April 15, 2014. There is no registration fee but we need your information, including your dates of arrival and departure, so that we may more effectively budget support and arrange housing.

Participants who plan to attend the conference should arrive on the evening of Wednesday May 14. The workshop will dismiss by Friday at 12:00 Noon. We expect that domestic participants  will check out of conference housing on Friday, May 23. Participants coming from abroad will be given housing for the evening of Friday May 23. Please contact the organizers if you have any questions. 

All those seeking support for travel, especially international visitors, should read carefully about the University of Kentucky's documentation requirements.  Please see the travel and accomodation page for more details.

The conference (May 15-17) will include tutorial lectures by the following researchers:

  • Kari Astala (University of Helsinki): Introduction to the functional-analytic theory of dbar-problems and other elliptic equations.
  • James Colliander (University of Toronto): Introduction to dispersive methods in partial differential equations.
  • Ken McLaughlin (University of Arizona): Introduction to orthogonal polynomials and random matrix theory
  • Peter Miller (University of Michigan): Introduction to semiclassical asymptotic analysis.
  • Andreas Stahel (Colorado State University, Fort Collins): Introduction to numerical methods for d-bar problems and other elliptic equations.
  • Peter Perry (University of Kentucky): The Davey-Stewartson II equation: a completely integrable model equation
  • Paolo Santini (University of Roma "La Sapienza"): Integrable systems and soliton Equations in 2+1 dimensions
  • Jean-Claude Saut (Universite de Paris-Sud): Partial differential equation methods versus integrable methods for dispersive nonlinear waves

The workshop (May 19-23) will bring together teams of senior scientists, postdoctoral researchers, and graduate students to work on unsolved problems in the field, including the following:

Semiclassical Limits. We will study semiclassical limits of direct and inverse scattering transforms for 2+1 dimensional problems. Such singular limits of nonlinear wave equations typically lead to a simplification of the dynamics in which different wave motions are confide to separate regions of the space-time, separated by asymptotically well-defined caustic curves.

Eigenvalue distributions of Random Normal Matrices. The analytical problem of large-N asymptotic behavior of random normal matrices has many features in common with semiclassical limits for dispersive nonlinear wave equations. Although some progress has been made for special potentials V which admit a reduction to a Riemann-Hilbert problem, it remains to understand how this analysis might inform a more general method not requiring special properties of the potential.

Direct Scattering and Exceptional Sets. Non-uniqueness in the multi-dimensional inverse scattering problem that define the scattering data lead to singularities in the scattering data and to such interesting physical phenomena as lump solutions, line solutions, and solutions of the Cauchy problem that blow up in finite time. We seek to clarify and classify the kinds of exceptional points that occur, and to elucidate their consequences for the associated dynamics.

Inverse Scattering and Exceptional Sets. One can also approach the problem of exceptional sets by beginning with a scattering transform with prescribed singularities, and reconstructing the potential from which it comes. Beginning with known examples, we seek to obtain sound theory and efficient algorithms for incorporating these singularities into the inverse scattering transform.

One-Dimensional Limits. 2+1 dimensional dispersive equations such as the Davey-Stewartson II equation have one-dimensional partners (the NLS, in case of DS II) to which they reduce for solutions independent of one of the spatial variables. We seek to understand the relationship between inverse scattering in one- and two-dimensions in order to elucidate two-dimensional phenomena such as line solitons that arise from one-dimensional soliton solutions but do not fit within the usual framework of inverse scattering theory.

For further information, please contact the organizers:

Kenneth T.-R. McLaughlin, mcl@math.arizona.edu Peter D. Miller, millerpd@umich.edu Peter A. Perry, peter.perry@uky.edu

We thank our sponsors:

 

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