English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We complete the program of Ref. [3,5] of connecting totally non-negativeGrassmannians to the reality problem in KP finite-gap theory via the assignmentof real regular divisors on rational degenerations of M-curves for the class ofreal regular multi-line soliton solutions of KP II equation whose asymptoticbehavior has been combinatorially characterized by Chakravarthy, Kodama andWilliams. We use the plabic networks in the disk introduced by Postnikov toparametrize positroid cells in totally nonnegative Grassmannians. The boundaryof the disk corresponds to the rational curve associated to the soliton data inthe direct spectral problem, and the bicolored graph is the dual of a reduciblecurve G which is the rational degeneration of a regular M-curve whose genus gequals the number of faces of the network diminished by one. We assign systemsof edge vectors to the planar bicolored networks. The system of relationssatisfied by them has maximal rank and may be reformulated in the form of edgesignatures.. We prove that the components of the edge vectors are rational inthe edge weights with subtraction free denominators and provide their explicitexpressions in terms of conservative and edge flows. The edge vectors rule thevalue of the KP wave function at the double points of G, whereas the signaturesat the vertices rule the position of the divisor in the ovals. We prove thatthe divisor satisfies the conditions of Dubrovin-Natanzon for real finite-gapsolutions: there is exactly one divisor point in each oval except for the onecontaining the essential singularity of the wave function. The divisor may beexplicitly computed using the linear relations at the vertices of the network.We explain the role of moves and reductions in the transformation of both thecurve and the divisor for given soliton data, and apply our construction tosome examples.

KP theory, plabic networks in the disk and rational degenerations of M--curves