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

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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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