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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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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Bologna, Bologna, Italy

    Simonetta Abenda

  2. L.D.Landau Institute for Theoretical Physics, pr. Ak Semenova 1a, Chernogolovka, Russia, 142432

    Petr G. Grinevich

  3. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, GSP-1, 1 Leninskiye Gory, Main Building, Moscow, Russia, 119991

    Petr G. Grinevich

  4. Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, Russia, 141700

    Petr G. Grinevich

Authors
  1. Simonetta Abenda
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  2. Petr G. Grinevich
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Correspondence to Simonetta Abenda.

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Communicated by C. Schweigert

This work has been partially supported by PRIN “Teorie geometriche e analitiche dei sistemi Hamiltoniani in dimensioni finite e infinite”, by the project “EQUATIONS”, by the Russian Foundation for Basic Research, Grant 17-01-00366, by the program “Fundamental problems of nonlinear dynamics”, Presidium of RAS.

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Abenda, S., Grinevich, P.G. Rational Degenerations of \({{\mathtt{M}}}\)-Curves, Totally Positive Grassmannians and KP2-Solitons. Commun. Math. Phys. 361, 1029–1081 (2018). https://doi.org/10.1007/s00220-018-3123-y

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