2012/05/05 by Yuji Kodama, Lauren Williams, Kodama, Yuji +1
Mathematics · Physics and Astronomy · #05Exx #13F60 #14P10 #14T05 #37K10 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1205.1101
openalex publication_date 2012/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 give an overview of our work on the combinatorics of such contour plots. Using the positroid stratification and the Deodhar decomposition of the Grassmannian (and in particular the combinatorics of Go-diagrams), we completely describe the asymptotics of these contour plots when |y| or |t| go to infinity. Other highlights include: a surprising connection with total positivity and cluster algebras; results on the inverse problem; and the characterization of regular soliton solutions -- that is, 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.