KP theory, planar bipartite networks in the disk and rational degenerations of M-curves

We associate real algebraic-geometric data ? la Krichever to any real regular multiline soliton solution of the Kadomtsev-Petviashvili II (KP) equation. These solutions correspond to a certain finite-dimensional reduction of the Sato Grassmannian and their asymptotic behavior is known to be classified in terms of the combinatorial structure of the totally non-negative part of real Grassmannians $Gr^{TNN} (k,n)$. In our construction, to any network representing a given soliton data in $Gr^{TNN} (k,n)$, we uniquely associate a rational degeneration of an M–curve of genus g and a real and regular degree g KP divisor on it. In particular, if we use the Le-network, then g is minimal and is equal to the dimension of the positroid cell to which the soliton datum belongs (joint research project with P.G. Grinevich (LITP, RAS))