Rational Degenerations of \({{\mathtt{M}}}\)-Curves, Totally Positive Grassmannians and KP2-Solitons
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Abstract
We establish a new connection between the theory of totally positive Grassmannians and the theory of \({{\mathtt{M}}}\)-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 \({Gr^{\textsc{tp}} (N,M)}\) a reducible curve which is a rational degeneration of an \({{\mathtt{M}}}\)-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 \({{\mathtt{M}}}\)-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 \({Gr^{\textsc{tp}} (r+1,M-N+r+1)\mapsto Gr^{\textsc{tp}} (r,M-N+r)}\).
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