Rational degeneration of M-curves, totally positive Grassmannians and KP2-solitons

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, 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 $G{r}^{TP}(N,M)$ 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 $M$-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 $G{r}^{TP}(r+1,M-N+r+1)\mapsto G{r}^{TP}(r,M-N+r)$.

49 pages, 10 figures. Minor revisions

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• On Reducible Degeneration of Hyperelliptic Curves and Soliton Solutions
• The direct scattering problem for perturbed Kadomtsev-Petviashvili multi line solitons
• Kasteleyn theorem, geometric signatures and KP-II divisors on planar bipartite networks in the disk
• Two-dimensional reductions of the Whitham modulation system for the Kadomtsev-Petviashvili equation