Solitons, Singularities, Surreals and Such: A Conference in Honor of Martin Kruskal's Eightieth Birthday

    Titles and Abstracts
    M. Ablowitz, University of Colorado
    "Solitary waves from optics to water waves"

    Abstract: Optical communications involves quite naturally the study of nonlinear waves. Asymptotic analysis leads to "classical", and "dispersion managed-nonlocal nonlinear Schrodinger equations which contain solitary waves as special solutions. A numerical method is introduced to find these and other solitary waves in optics. In water waves the classical equations are reformulated as integral equations. Asymptotic reductions of this syetems yields the Boussinesq/KdV, Benney-Luke/Kadomtsev-Petviashvili and nonlinear Schr\"odinger equations. The above numerical method is extended to find one dimensional solitary waves and two dimensional lumps with sufficient surface tension.

    F. Calogero, University of Rome
    "The Transition from Regular to Irregular Motions, Explained as Travel on Riemann Surfaces"

    Abstract: This is joint work with David Gomez-Ullate, Paolo Santini and Matteo Sommacal. We introduce and discuss a simple Hamiltonian dynamical system, interpretable as a 3-body problem in the (complex) plane and providing the prototype of a mechanism explaining the transition from regular to irregular motions as travel on Riemann surfaces. The interest of this phenomenology -- illustrating the onset in a deterministic context of irregular motions -- is underlined by its generality, suggesting its eventual relevance to understand natural phenomena and experimental investigations. The interest of this toy model is that, even though some of its motions can justifiably be considered "chaotic", its time evolution can nevertheless be described rather precisely and explicitly (thereby "predicting the unpredictable"!). The origin of the mechanism illuminated by this model is based on ideas pioneered by Martin Kruskal.

    P. A. Clarkson, Kent University, UK
    "Special Polynomials Associated with Rational Solutions of the Painleve Equations"

    Abstract: Please click here to view abstract.

    O. Costin, Ohio State University
    "Discrete Painlev\'e Property, and Integrability versus Chaos in Difference Equations"

    Abstract:Generalized Borel summability is used to analyze a relatively general class of difference equations. Their solutions are continued in a unique way in the complex $n$ domain. This continuation allows for the Painlev\e property (PP), one of the most effective indicators of integrability, to be extended to difference equations. It is shown that the PP induces, under relatively general assumptions, a dichotomy within first order difference equations: all equations with the PP can be solved in closed form; on the contrary, absence of the PP implies, under some further assumptions, that locally conserved quantities are in fact strictly local, in the sense that they develop singularity barriers on the boundary of some compact set. The techniques allow for a detailed description of Julia sets of maps of the interval. Higher order equations will be discussed as well.
    Based on work in collaboration with Martin Kruskal, to appear in Comm. Pure Appl. Math.

    R. D. Costin, Ohio State University
    "Integrability, Nonintegrability and the Poly-Painleve Test"

    Abstract: One of the fundamental questions in the theory of differential equations is to determine whether an equation is integrable or not. ``When, however, one attempts to formulate a precise definition of integrability, many possibilities appear[...]'' (D. Birkhoff)

    Proposing a general, comprehensive point of view on integrability, Kruskal has devised a way of testing for this property: the Poly-Painleve test. The main ideas are presented, as well as recent developments stemming from it: rigorous integrability criteria for classes of equations, regularity of first integrals, classification of equations in singular regions.

    P. Deift, New York University
    "Universality for Orthogonal and Symplectic Ensembles in Random Matrix Theory"

    Abstract:The speaker will describe recent results on universality in the bulk and at the edge for orthogonal and symplectic random matrix ensembles. The proofs utilize formulae of Tracy/Widom and Widom, together with Riemann-Hilbert/steepest descent methods.
    This is joint work with Dimitri Gioev.

    N. Joshi, University of Sydney Australia
    "Analytic results for (ultra-discrete) cellular automata"

    Abstract:Cellular automata (CA) have been widely adopted in the sciences as simple but powerful models of the real world. The complex patterns produced by their long-time behaviours are used to confirm the correctness of scientific hypotheses underlying the model. However, the analysis essential for testing scientific hypotheses appears to be missing. We show the beginnings of the mathematical analysis needed by reviewing the connection that leads from differential equations to ultra-discrete equations which are extended CA. New results for ultra-discrete equations include a test for integrability and results on solvability through associated linear problems.

    C. P. Kruskal, University of Maryland
    "The Chromatic Number of the Plane: the Bounded Case"

    R. Kulsrud, Princeton University
    "Scaling Matterhorn with Martin in the Fifties"

    P. Lax, New York University
    "The Zero Dispersion Limit for KdV"

    R. M. Miura, New Jersey Institute of Technology
    "Formation of Glass Microelectrodes"

    S. Orszag, Yale University
    "Asymptotic Analysis of the Extreme Statistical Mechanics of Networked Computer Storage "

    A. Ramani, Ecole Polytechnique, France
    "What Is It that Mathematicains Do and Computers Cannot"

    H. Segur, University of Colorado at Boulder
    "Soliton models of waves in shallow water"
    Abstract: The theory of solitons and of completely integrable problems began in 1965, when Zabusky, Kruskal and their co-workers showed that the Korteweg-de Vries (KdV) equation has extraordinary mathematical structure. The Kadomtsev-Petviashivili (KP) equation, which was constructed in 1970 to generalize the KdV equation to two spatial dimensions plus time, also has extraordinary mathematical structure: both equations are completely integrable. In addition, both equations describe approximately the evolution of waves in shallow water, and neither set of authors knew that their equation had extraordinary structure when they derived it as a mathematical model of a physical problem. This talk reviews some aspects of how these two completely integrable models (KdV and KP) have changed our understanding of the dynamics of actual waves in shallow water.

    E. Witten, Institute for Advanced Study
    ``Gauge Theory And The Geometric Langlands Program.''

    I will explain how four-dimensional gauge theory with electric and magnetic charges can be applied to a mathematical theory known as the geometric Langlands program (no prior knowledge of which is assumed).