Rational Degenerations of M-Curves, Totally Positive Grassmannians and KP2-Solitons

We establish a new connection between the theory of totally positive Grassmannians and the theory of M-curves using the finite-gap theory for solitons of the KP equation. Here and in the following KP equation denotes the Kadomtsev–Petviashvili 2 equation [see (1)], which is the first flow from the KP hierarchy. We also assume that all KP times are real. We associate to any point of the real totally positive Grassmannian GrTP(N,M) a reducible curve which is a rational degeneration of an M-curve of minimal genus g=N(M−N), and we reconstruct the real algebraic-geometric data á la Krichever for the underlying real bounded multiline KP soliton solutions. From this construction, it follows that these multiline solitons can be explicitly obtained by degenerating regular real finite-gap solutions corresponding to smooth M-curves. In our approach, we rule the addition of each new rational component to the spectral curve via an elementary Darboux transformation which corresponds to a section of a specific projection GrTP(r+1,M−N+r+1)↦GrTP(r,M−N+r)