Yakar Kannai, Weizmann Institute of Science
Explicit solutions for 2nd order degenerate equations

Exponentials solve differential equations with constant coefficients, leading to Fourier Transform and to ψ do's. The challenging degenerate (sum of squares) case leads to Sub-Riemannian geometry and complex Hamiltonians in the step two case. Special functions surface in the higher step low dimensional case.

Joel Lebowitz, Rutgers University
Microscopic derivation of driven reaction-diffusion equations with boundaries

We study interacting particle (spin) systems on a lattice under the combined influence of Glauber (spin flip) and simple exchange (Kawasaki) dynamics. We prove that when the conserving exchanges occur on a microscopically fast scale the macroscopic density (magnetization) evolves according to nonlinear drift-diffusion-reaction equation. Microscopic fluctuations about the deterministic macroscopic evolution are also found explicitly. They grow, with time, to become infinite, when the deterministic solution is unstable.

Boris Levant, Weizmann Institute of Science
Shell models as phenomenological models of turbulence: Theory and computations

In the last decade shell models of turbulence have attracted a lot of attention in the physical community. Many numerical studies have confirmed, that although the models are much simpler than the original Navier-Stokes equations, they contain certain features of the real world turbulence. In my talk I will introduce the models and describe a few recent analytic results. Specifically, I will concentrate on the two most popular shell models GOY and Sabra, and describe my results concerning regularity of their solutions and finite dimensionality of their long-time behavior.

I will give further motivation for the analytic study of the shell models. In particular, I will discuss one of the most challenging problems in the theory of turbulence - a problem of anomalous scaling of the velocity structure functions. I will present analytic and numerical results to show how shell models were used in trying to solve this question.

In the talk I will present results of the joint works with R. Benzi, P. Constantin, I. Procaccia and E. S. Titi.

Ehud Meron, Ben Gurion University
Nonlinear physics of water-limited ecosystems

The impacts of global climate change and human activities on the stability of marine and terrestrial ecosystems pose great challenges for current research in ecology. Of particular concern are processes involving desertification and biodiversity change. In this talk I will describe recent model studies that shed new light on the mechanisms involved in such processes. The mechanisms are intimately related to concepts of nonlinear physics such as spatial symmetry breaking, multiplicity of stable states, front propagation and collective dynamics. Processes to be addressed include vegetation pattern formation along rainfall gradients, shrubland-grassland transitions, and ecosystem engeneering leading to fertility islands in stressed environments.

Jeremy Schiff, Bar Ilan University
Painleve tests and normal forms at infinity

Painleve tests are usually thought of as a set of methods for detecting integrability of a (system of) differential equation(s). The methods involve checking the consistency of various kinds of formal series solution for the equation, and in particular trying to establish the absence of a certain kind of logarithmic singularity. The rough idea behind this is that such singularities give rise to complex, chaotic behavior, and thus integrable systems cannot have them.

In this talk I will review some of the rigorous results that exist relating Painleve tests to the existence of conserved quantities. In 2000 Goriely proved that most Painleve tests of a system of ODEs are equivalent to constructing the normal form for certain "companion systems" of ODEs near a fixed point. I will show that Goriely's companion systems can be replaced by other systems that arise directly from changes of coordinate in the original system. The fixed points of the companion systems correspond to points at infinity in the original coordinate systems, and thus Painleve tests can, and should, be thought of, as constructing normals form at infinity for a system of equations.

It is thus to be expected that Painleve tests should have applications well beyond just integrable systems. In this talk, however, I will focus on how the new understanding resolves some outstanding issues of how to correctly apply Painleve tests, and why they are unfortunately very limited.

Steve Schochet, Tel Aviv University
Multiple-time-scale expansions for very long times

Multiple-time-scale expansions are a flexible tool for determining how solutions to differential equations depend on a small parameter in those equations. Two-time-scale expansions are equivalent to the method of averaging and to numerous other methods. Textbook wisdom says that three-time-scale expansions can be obtained only in trivial cases when one of the scales turns out to be absent from the solution. Indeed, examples have been obtained that illustrate the failure of three-time-scale expansions. Nevertheless, an appropriate reformulation of the multiple-time-scale method yields expansions for some of those examples and a certain class of problems. However, the scope of the method is inherently limited to problems for which perturbations do not grow exponentially.

Yossi Shamai, Hebrew University
Spatial correlations and variance reduction in Brownian simulations

Brownian simulation methods have become very popular in the context of polymeric fluids since the 1990's. In this lecture, we will present a general formulation of error minimization and optimal optimal noise correlation. We will show that the resulting optimization problem is amenable to an algorithmic solution. Specific examples will be considered and general considerations will be discussed.

Marshall Slemrod, University of Wisconsin
Steady Transonic Flow

On two significant papers written a decade apart Cathleen Morawetz ( 1985,1995) showed that the Murat-Tartar theory of compensated compactness might be amenable to proving the existence of weak solutions to the equations of two dimensional steady transonic gas dynamics. Morawetz assumed the existence of an unspecified "viscous" approximating system that would provide the desired estimates for the compensated compactness theory to apply. In this talk I will describe recent work by G Q Chen, C M Dafermos, M Slemrod ( CMP. 2007) and G Q Chen, M Slemrod, D Wang ( ARMA, 2007) which provides such a viscous formulation. The approach also suggests a new method for numerical solution of transonic gas dynamics as well as resolving problems in isometric embedding of two dimensional Riemannian manifolds in the three dimensional Euclidean space with ( smooth) Gauss curvature changing sign.

Irad Yavneh, Technion
An adaptive multilevel approach to eigenproblems with application to PageRank

* Joint work with Yair Koren, Achi Brandt and Panayot Vassilevski

A new multilevel approach is presented for the numerical solution of eigenvalue problems. As in classical multigrid methods, the algorithm alternates between a simple iteration, such as some variant of the power method, and a coarsening process, whereby the problem is approximated using a smaller matrix. This is applied recursively to produce a highly efficient solver for certain types of sparse matrices. The novelty of the method is manifest in the nonlinear adaptive manner of constructing the smaller matrix, such that the original problem is represented more and more accurately by the small problem as the algorithm progress.

The talk will describe work in progress. The motivating problem is the celebrated PageRank problem introduced by Google to rank web pages according to "importance". The problem will be described and examples using actual segments of the web will be tested.