Location: 384I.
Time: Monday, 4pm.
Abstract: This talk reports in part on joint work with Russell Brown (Kentucky), Katherine Ott (Bates College), Michael Music (Kentucky), Jiaqi Liu (Kentucky), and Catherine Sulem (Toronto).
In recent years, inverse scattering theory (IST) has been developed into a mathematically rigorous and effective tool for studying global well-posedness and long-time behavior of completely integrable dispersive PDE's in one and two dimensions. I will discuss Deift and Zhou's pioneering work on the cubic NLS in one dimension (2003) and more recent work:
(1) (Liu, Perry, Sulem) the derivative nonlinear Schrodinger equation, a 1+1 dimensional equation that models the dynamics of Alfven waves propagating along an ambient magnetic field in a long-wave, weakly nonlinear scaling regime;
(2) (Perry) the Davey-Stewartson equation, a dispersive equation in two dimensions that describes modulated nonlinear surface gravity waves propagating over a horizontal sea bed, and
(3) (Music, Perry) the Novikov-Veselov equation, a two-space dimensional analogue of the celebrated Korteweg-de Vries equation that includes the celebrated Kadomtsev-Petviashvili equations as scaling limits.
I will also discuss prospects for further progress on long-term behavior for integrable dispersive PDE's and small perturbations from integrability.
References:
(1) P. Deift, X. Zhou. Long-time asymptotics for solutions of the NLS equation with initial data in a weighted Sobolev space. Dedicated to the memory of Jürgen K. Moser. Comm. Pure Appl. Math. 56 (2003), no. 8, 1029-1077
(2) P. Perry, Global Well-Posedness and Long-time Asymptotics for the Defocussing Davey-Stewartson II Equation in $H^{1,1}(R^2)$ arXiv:1110.5589v3 (Accepted pending revisions for J. Spectral Theory).
(3) P.A. Perry, K.A. Ott, R.M. Brown. Action of a scattering map on weighted Sobolev spaces in the plane
arXiv:1501.04669 (submitted to J. Funct. Anal.)
(4) Mi. Music, P. A. Perry. Global solutions for the zero-energy Novikov-Veselov equation by inverse scattering.
arXiv:1502.02632 (submitted to Comm. Math. Phys.)
(5) J. Liu, P. A. Perry, C. Sulem. Global existence for the Derivative Nonlinear Schrodinger Equation by the method of inverse scattering. To be posted on arXiv.