Lakeside Lectures

Speaker:  Prof. Yuji Kodama(Ohio State University)
 
Title:  Combinatorics and geometry of KP solitons and applications to tsunami
 
Abstract:  Let Gr(N,M) be the real Grassmann variety defined by the set of all N-dimensional subspaces of R^M.  Each point on Gr(N,M) can be represented by an NxM matrix A of rank N. If all the NxN 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)^+. In this talk, I start to give a realization of Gr(N,M)^+ in terms of the (regular) soliton solutions of the KP (Kadomtsev-Petviashvili) 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)^+ 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 Le-diagram provides a complete classification of the ''entire'' spatial patterns of the soliton solutions coming from the Gr(N,M)^+ 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.
 
Date:  Mar 16th, 2015
 
Time: 14:00-15:00
 
Venue: Room 202,2/F, Astro-Math. Building
 
Refreshment:13:30-14:00
 
Organizers: Chin-Yu Hsiao, Mao-Pei Tsu, Fu-Tsun Wei, Jeng-Daw Yu