General Information

The Equadiff 07 conference takes place at the Vienna University of Technology, August 5-11, 2007. Further information can be found on the official website. The minisymposium on Integrable Systems is organised by Spyridon Kamvissis (Crete) and Johanna Michor (London).

Timetable

Tuesday, August 7, 13:30h-15:30h and 16h-18:30h, lecture hall 8.

Time	Tuesday, August 7
11:20- 12:05	Stephanos Venakides
13:30	Christian Klein
14:00	Tom Claeys
14:30	Marco Bertola
15:00	Vassilis Papanicolaou
15:30
16:00	Johanna Michor
16:30	Peter Yuditskii
17:00	Spyridon Kamvissis

List of talks

The following participant will give a 45 minutes talk.

Stephanos Venakides, Duke	The focusing nonlinear Schrödinger (NLS) equation: Rigorous semiclassical and long time asymptotics
	Abstract in PDF-format.

The following participants will give a 25 minutes talk.

Marco Bertola, Montreal	Boutroux curves with external potential: equilibrium measures without a minimization problem
	Abstract. The nonlinear steepest descent method for rank-two systems relies on the notion of g-function. For the case of asymptotics of generalized orthogonal polynomials with respect to varying complex weights we can recast those requirements in a problem in algebraic geometry and harmonic analysis and completely solve the existence and uniqueness issue without relying on the minimization of a functional. This addresses and solves also the issue of the "free boundary problem", determining implicitly the curves where the zeroes of the orthogonal polynomials accumulate in the limit of large degrees. The notion and techniques developed here are not limited to (pseudo) orthogonal polynomials but extend to other settings where the nonlinear steepest descent method is used including Painleve equations. The sudden topological changes (w.r.t. parameters) of the structure of the "free boundary" hinted at earlier are the essence of the so-called "nonlinear Stokes' phenomenon".
Tom Claeys, Leuven	A real pole-free solution to a higher order Painleve I equation
	Abstract. We consider a fourth order ODE, which is an analogue to the Painleve I equation. We prove the existence of a real pole-free solution to this equation, which was conjectured by Dubrovin, using Riemann-Hilbert methods. This existence follows from the solvability of a certain Riemann-Hilbert problem, which can be proven using a so-called vanishing lemma. The pole-free solution plays a role in critical random matrix ensembles, and describes asymptotics in the small dispersion limit of the KdV equation near the point of gradient catastrophe. The talk is based on joint work with Maarten Vanlessen.
Spyridon Kamvissis, Crete	Stability of the periodic Toda lattice
	Abstract. We consider the stability of the periodic Toda lattice (and slightly more generally of the algebro-geometric finite-gap lattice) under a short range perturbation. We prove that the perturbed lattice asymptotically approaches a modulated lattice. Our method relies on the equivalence of the inverse spectral problem to a matrix Riemann-Hilbert problem defined on the hyperelliptic curve and generalizes the so-called nonlinear stationary phase/steepest descent method for Riemann-Hilbert problem deformations to Riemann surfaces. This is work done with Gerald Teschl.
Christian Klein, Tübingen	Dissipationless shocks and Painleve equations
	Abstract. The Cauchy problem for dissipationless equations as the Korteweg de Vries (KdV) equation with small dispersion of order ε^2, ε << 1, is characterized by the appearance of a zone of rapid modulated oscillations of wave-length of order ε. Near the gradient catastrophe of the dispersionless equation (ε=0), a multi-scales expansion gives an asymptotic solution in terms of a fourth order generalization of Painleve I. At the leading edge of the oscillatory zone, a corresponding multi-scales expansion yields an asymptotic description of the oscillations where the envelope is given by a solution to the Painleve II equation. We study the applicability of these approximations for several PDEs numerically.
Johanna Michor, London	Algebro-geometric finite-band solutions of the Ablowitz-Ladik hierarchy
	Abstract. We will discuss a detailed derivation of all complex-valued algebro-geometric finite-band solutions of the Ablowitz-Ladik hierarchy, a completely integrable sequence of systems of nonlinear evolution equations on the lattice Z. In addition, we survey a recursive construction of the Ablowitz-Ladik hierarchy and its zero-curvature and Lax formalism. The talk is based on joint work with Fritz Gesztesy, Helge Holden, and Gerald Teschl.
Vassilis Papanicolaou, Athens	Higher-Order Periodic Operators
	Abstract. We discuss the periodic spectral and inverse spectral theory of higher-order ordinary differential operators including some basic, but still unanswered questions. This theory has interesting analytic and algebrogeometric aspects. We follow the plan of Prof. S. Novikov. Our "guide" is the Euler-Bernoulli operator whose theory is now completely understood.
Peter Yuditskii, Linz	The "action" variable is not an invariant for the uniqueness in the inverse scattering problem
	Abstract. We give a simple example of non-uniqueness in the inverse scattering for Jacobi matrices: roughly speaking S-matrix is analytic. Then, multiplying a reflection coefficient by an inner function, we repair this matrix in such a way that it does uniquely determine a Jacobi matrix of Szegö class; on the other hand the transmission coefficient remains the same. This implies the statement given in the title.

Abstracts

Abstracts can be found here. All participants should submit their abstract before June 30th on the official website.

Registration, travel, and accommodation

Every participant is responsible for registering with the conference and for her/his travel and accommodation. Further information is available at the official website.

Hope to see you in Vienna! Spyros and Johanna.