Colloque

13 décembre 2013 de 16 h 00 à 18 h 00 (heure de Montréal/HNE) Sur place

Combinatorics and geometry of KP solitons and application to tsunami

Let $Gr(N,M)$ be the real Grassmann manifold defined by the set of all $N$-dimensional subspaces of ${\mathbb R}^M$. Each point on $Gr(N,M)$ can be represented by an $N\times M$ matrix $A$ of rank $N$. If all the $N\times N$ minors of $A$ are nonnegative, the set of all points associated with those matrices forms the totally nonnegative part of the Grassmannian, denoted by $Gr(N,M)_{\ge 0}$. In this talk, I start to give a realization of $Gr(N,M)_{\ge 0}$ in terms of the (regular) soliton solutions of the KP (KadomtsevPetviashvili) equation which is a two-dimensional extension of the KdV equation. The KP equation describes small amplitude and long waves on a surface of shallow water. I then construct a cellular decomposition of $Gr(N,M)_{\ge 0}$ with the asymptotic form of the soliton solutions. This leads to a classification theorem of all solitons solutions of the KP equation, showing that each soliton solution is uniquely parametrized by a derrangement of the symmetric group $S_M$. Each derangement defines a combinatorial object called the Le-diagram (a Young diagram with zeros in particular boxes). Then I show that the Lediagram provides a complete classification of the ''entire'' spatial patterns of the soliton solutions coming from the $Gr(N,M)_{\ge 0}$ for asymptotic values of the time. I will also present some movies of real experiments of shallow water waves which represent some of those solutions obtained in the classification problem. Finally I will discuss an application of those results to analyze the Tohoku-tsunami on March 2011. The talk is elementary, and shows interesting connections among combinatorics, geometry and integrable systems.