The Deodhar decomposition of the Grassmannian and the regularity of KP solitons

Abstract

Given a point A in the real Grassmannian, it is well-known that one can construct a soliton solution uA(x,y,t) to the KP equation. The contour plot of such a solution provides a tropical approximation to the solution when the variables x, y, and t are considered on a large scale and the time t is fixed. In this paper we use several decompositions of the Grassmannian in order to gain an understanding of the contour plots of the corresponding soliton solutions. First we use the positroid stratification of the real Grassmannian in order to characterize the unbounded line-solitons in the contour plots at y≫0 and y≪0. Next we use the Deodhar decomposition of the Grassmannian–a refinement of the positroid stratification–to study contour plots at t≪0. More specifically, we index the components of the Deodhar decomposition of the Grassmannian by certain tableaux which we call Go-diagrams, and then use these Go-diagrams to characterize the contour plots of solitons solutions when t≪0. Finally we use these results to show that a soliton solution uA(x,y,t) is regular for all times t if and only if A comes from the totally non-negative part of the Grassmannian.