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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 Schroedinger (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.
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 Schroedinger (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.
This book describes the theory and applications of discrete
orthogonal polynomials--polynomials that are orthogonal on a finite
set. Unlike other books, "Discrete Orthogonal Polynomials"
addresses completely general weight functions and presents a new
methodology for handling the discrete weights case.
J. Baik, T. Kriecherbauer, K. T.-R. McLaughlin & P. D.
Miller focus on asymptotic aspects of general, nonclassical
discrete orthogonal polynomials and set out applications of current
interest. Topics covered include the probability theory of discrete
orthogonal polynomial ensembles and the continuum limit of the Toda
lattice. The primary concern throughout is the asymptotic behavior
of discrete orthogonal polynomials for general, nonclassical
measures, in the joint limit where the degree increases as some
fraction of the total number of points of collocation. The book
formulates the orthogonality conditions defining these polynomials
as a kind of Riemann-Hilbert problem and then generalizes the
steepest descent method for such a problem to carry out the
necessary asymptotic analysis.
This book represents the first asymptotic analysis, via
completely integrable techniques, of the initial value problem for
the focusing nonlinear Schrodinger equation in the semiclassical
asymptotic regime. This problem is a key model in nonlinear optical
physics and has increasingly important applications in the
telecommunications industry. The authors exploit complete
integrability to establish pointwise asymptotics for this problem's
solution in the semiclassical regime and explicit integration for
the underlying nonlinear, elliptic, partial differential equations
suspected of governing the semiclassical behavior. In doing so they
also aim to explain the observed gradient catastrophe for the
underlying nonlinear elliptic partial differential equations, and
to set forth a detailed, pointwise asymptotic description of the
violent oscillations that emerge following the gradient
catastrophe.
To achieve this, the authors have extended the reach of two
powerful analytical techniques that have arisen through the
asymptotic analysis of integrable systems: the
Lax-Levermore-Venakides variational approach to singular limits in
integrable systems, and Deift and Zhou's nonlinear
Steepest-Descent/Stationary Phase method for the analysis of
Riemann-Hilbert problems. In particular, they introduce a
systematic procedure for handling certain Riemann-Hilbert problems
with poles accumulating on curves in the plane. This book, which
includes an appendix on the use of the Fredholm theory for
Riemann-Hilbert problems in the Holder class, is intended for
researchers and graduate students of applied mathematics and
analysis, especially those with an interest in integrable systems,
nonlinear waves, or complex analysis."
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