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Reducible M-curves for Le-networks in the totally-nonnegative Grassmannian and KP-II multiline solitons

  • Simonetta AbendaEmail author
  • Petr G. Grinevich
Article
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Abstract

We associate real and regular algebraic–geometric data to each multi-line soliton solution of Kadomtsev–Petviashvili II (KP) equation. These solutions are known to be parametrized by points of the totally non-negative part of real Grassmannians \(Gr^{\mathrm{TNN}}(k,n)\). In [3] we were able to construct real algebraic–geometric data for soliton data in the main cell \(Gr^{\mathrm{TP}} (k,n)\) only. Here we do not just extend that construction to all points in \(Gr^{\mathrm{TNN}}(k,n)\), but we also considerably simplify it, since both the reducible rational \(\texttt {M}\)-curve \(\Gamma \) and the real regular KP divisor on \(\Gamma \) are directly related to the parametrization of positroid cells in \(Gr^{\mathrm{TNN}}(k,n)\) via the Le-networks introduced in [63]. In particular, the direct relation of our construction to the Le-networks guarantees that the genus of the underlying smooth \(\texttt {M}\)-curve is minimal and it coincides with the dimension of the positroid cell in \(Gr^{\mathrm{TNN}}(k,n)\) to which the soliton data belong to. Finally, we apply our construction to soliton data in \(Gr^{\mathrm{TP}}(2,4)\) and we compare it with that in [3].

Keywords

Total positivity Totally non-negative Grassmannians KP hierarchy Real solitons M-curves Le-diagrams Planar bicolored networks in the disk Baker–Akhiezer function 

Mathematics Subject Classification

37K40 37K20 14H50 14H70 

Notes

Acknowledgements

The authors would like to express their gratitude to the referee for careful reading of the manuscript and useful remarks.

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Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di BolognaBolognaItaly
  2. 2.L.D. Landau Institute for Theoretical Physics, RASChernogolovkaRussia
  3. 3.Faculty of Mechanics and MathematicsLomonosov Moscow State UniversityMoscowRussia
  4. 4.Moscow Institute of Physics and TechnologyDolgoprudnyRussia

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