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

@article{Abenda2018ReducibleMF,
title={Reducible M-curves for Le-networks in the totally-nonnegative Grassmannian and KP-II multiline solitons},
author={Simonetta Abenda and P. G. Grinevich},
journal={Selecta Mathematica},
year={2018},
volume={25},
pages={1-64}
}
• Published 15 May 2018
• Mathematics
• Selecta Mathematica
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 $$Gr^{\mathrm{TNN}}(k,n)$$GrTNN(k,n). In [3] we were able to construct real algebraic–geometric data for soliton data in the main cell $$Gr^{\mathrm{TP}} (k,n)$$GrTP(k,n) only. Here we do not just extend that construction to all points in $$Gr^{\mathrm… 15 Citations • Mathematics Letters in Mathematical Physics • 2022 In this paper, we construct an explicit map from planar bicolored (plabic) trivalent graphs representing a given irreducible positroid cell$${{\mathcal {S}}}_{{\mathcal {M}}}^{\text{ TNN }}$$S • Mathematics • 2020 In this paper we use the fact that every Postnikov planar bicolored (plabic) trivalent graph representing a given irreducible positroid cell S in the totally non-negative Grassmannian In this paper, we construct an explicit map from planar bicolored (plabic) trivalent graphs representing a given irreducible positroid cellSTNN M in the totally non-negative Grassmannian GrTNN(k, n) • Mathematics • 2017 We complete the program of Ref. [3,5] of connecting totally non-negative Grassmannians to the reality problem in KP finite-gap theory via the assignment of real regular divisors on rational • Mathematics International Mathematics Research Notices • 2022 The standard parametrization of totally non-negative Grassmannians was obtained by A. Postnikov [45] introducing the boundary measurement map in terms of discrete path integration on planar • S. Abenda • Mathematics Mathematical Physics, Analysis and Geometry • 2021 Maximal minors of Kasteleyn sign matrices on planar bipartite graphs in the disk count dimer configurations with prescribed boundary conditions, and the weighted version of such matrices provides a • Mathematics • 2018 We apply the general construction developed in references [4,5] to the first non-trivial case of Gr^{TP}(2,4). In particular, we construct finite-gap KP-II real quasiperiodic solutions in the form of Following [43], positroid cells S M in totally non-negative Grassmannians Gr(k,n) admit parametrizations by positive weights on planar bicolored directed perfect networks in the disk. An explicit • Mathematics • 2021 Following [42], positroid cells S M in totally non-negative Grassmannians Gr(k,n) admit parametrizations by positive weights on planar bicolored directed perfect networks in the disk. An explicit ## References SHOWING 1-10 OF 117 REFERENCES We construct the pole divisor of the wavefunction for real regular bounded multi-soliton KP-II solutions represented by points in$$Gr^\mathrm{TP} (2,4)$$GrTP(2,4) on the reducible rational$$\mathtt
• Mathematics
• 2018
AbstractWe establish a new connection between the theory of totally positive Grassmannians and the theory of $${{\mathtt{M}}}$$M-curves using the finite-gap theory for solitons of the KP equation.
• Mathematics
• 2017
We complete the program of Ref. [3,5] of connecting totally non-negative Grassmannians to the reality problem in KP finite-gap theory via the assignment of real regular divisors on rational
• Mathematics
Moscow Mathematical Journal
• 2019
The family of complex Grassman manifolds $G_{n,k}$ with the canonical action of the torus $T^n=\mathbb{T}^{n}$ and the analogous of the moment map $\mu : G_{n,k}\to \Delta _{n,k}$ for the
• Mathematics
• 2009
In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian, denoted (Grk,n)≥0. This is a cell complex whose cells ΔG can be parameterized in
• Mathematics
• 2018
We apply the general construction developed in references [4,5] to the first non-trivial case of Gr^{TP}(2,4). In particular, we construct finite-gap KP-II real quasiperiodic solutions in the form of
AbstractWe describe a family of the rational solutions of the Zakharov—Schabat equations. This family is characterized by extremely simple superposition principle, following directly from the
• Mathematics
Functional Analysis and Geometry
• 2019
A rectangular matrix is called totally positive, if all its minors are positive. A point of a real Grassmanian manifold $G_{l,m}$ of $l$-dimensional subspaces in $\mathbb R^m$ is called strictly