Random solitons and soliton gasses

• Manuela Girotti (Emory University)

The concept of a soliton gas was originally introduced by Zakharov in the Seventies and it can be loosely described as a class of solutions of nonlinear integrable PDEs that displays an infinite number of -possibly random- solitons. I will present a collection of recent results from a collaborative effort to rigorously understand various aspects of soliton gas models: long-time asymptotics, random statistics, and interaction dynamics. In particular, we consider a gas of random $N$ solitons for the KdV equation and we study the limit as $N$ goes to infinity. We derive the limiting solution and we prove that its fluctuations are Gaussian random variables. The starting point is the formulation of a multi-soliton solution and of a regular, dense soliton gas solution in terms of Riemann–Hilbert problems. Tools from asymptotic analysis and probability will be used to provide a detailed description of interesting phenomena. This is a collection of works (and ongoing collaborations) done with K. McLaughlin (Tulane U.), T. Grava (SISSA/Bristol), R. Jenkins (UCF), A. Minakov (U. Karlova), J. Najnudel (Bristol).

Universal Asymptotics of Eulerian maps

• Nathan Hayford (University of Michigan)

The connection between matrix models and maps has been well-known and heavily exploited since its discovery in the late 70s. Most studies have been limited to regular maps, i.e. maps where all vertices have the same degree. This limitation was in part due to a substantial leap in difficulty in studying multivariate (as opposed to univariate) generating functions. Recently, we have been able to overcome this obstacle by introducing new techniques in the theory of analytic combinatorics in several variables. We are thus able to make statements about more general classes of maps. I will discuss these new techniques, and describe the universal appearance of Painlevé I predicted by Bender, Gao, and Richmond. This is joint work with Ahmad Barhoumi and Roozbeh Gharakhloo.