Chords and solitons: KP solitons in shallow water

Event Detail

Event type: Applied Mathematics and Computation Seminar
Date/Time: 09/15/2009-14:00   
Location: GLK 113
More info: http://www.math.oregonstate.edu/amc_seminar

Speaker info

Speaker: Yuji Kodama, Ohio State University

Abstract:
(Host: Prof. Harry Yeh) Let Gr$(N,M)$ be the real Grassmannian defined by the set of all $N$-dimenaional 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$^+(N,M)$.
In this talk, I will give a realization of Gr$^+(N,M)$ in terms of the soliton solutions of the KP equation, and 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 permutation group $S_M$. Expressing each derrangement by a unique chord diagram, I will show that the chord diagrams can be used to analyze the asymptotic behavior of certain initial value problems of the KP equation. I will also present some movies of real experiments of shallow water waves which represent some of new solutions obtained in the classification problem.