This volume contains lectures and invited papers from the Focus Program
on "Nonlinear Dispersive Partial Differential Equations and Inverse
Scattering" held at the Fields Institute from July 31-August 18, 2017.
The conference brought together researchers in completely integrable
systems and PDE with the goal of advancing the understanding of
qualitative and long-time behavior in dispersive nonlinear equations.
The program included Percy Deift's Coxeter lectures, which appear in
this volume together with tutorial lectures given during the first week
of the focus program. The research papers collected here include new
results on the focusing nonlinear Schrödinger (NLS) equation, the
massive Thirring model, and the Benjamin-Bona-Mahoney equation as
dispersive PDE in one space dimension, as well as the
Kadomtsev-Petviashvili II equation, the Zakharov-Kuznetsov equation, and
the Gross-Pitaevskii equation as dispersive PDE in two space dimensions.
The Focus Program coincided with the fiftieth anniversary of the
discovery by Gardner, Greene, Kruskal and Miura that the Korteweg-de
Vries (KdV) equation could be integrated by exploiting a remarkable
connection between KdV and the spectral theory of Schrodinger's equation
in one space dimension. This led to the discovery of a number of
completely integrable models of dispersive wave propagation, including
the cubic NLS equation, and the derivative NLS equation in one space
dimension and the Davey-Stewartson, Kadomtsev-Petviashvili and
Novikov-Veselov equations in two space dimensions. These models have
been extensively studied and, in some cases, the inverse scattering
theory has been put on rigorous footing. It has been used as a powerful
analytical tool to study global well-posedness and elucidate asymptotic
behavior of the solutions, including dispersion, soliton resolution, and
semiclassical limits.
