Rational degeneration of M-curves, totally positive Grassmannians and KP-solitons

The aim of our paper is to connect two areas of mathematics: -The theory of totally positive Grassmannians; -The rational degenerations of the $M$-curves; using the finite--gap theory for solitons of the Kadomtsev-Petviashvili 2 (KP) equation. We associate to any point of the real totally positive Grassmannian $Gr^{\mbox{\tiny TP}} (N,M)$ the rational degeneration of an M-curve of minimal genus g=N(M-N) and we reconstruct the algebro-geometric data a la Krichever for the underlying line soliton solutions to the KP equations.