KP theory, plabic networks in the disk and rational degenerations of Mcurves
Abstract
We complete the program of Ref. [3,5] of connecting totally nonnegative Grassmannians to the reality problem in KP finitegap theory via the assignment of real regular divisors on rational degenerations of Mcurves for the class of real regular multiline soliton solutions of KP II equation whose asymptotic behavior has been combinatorially characterized by Chakravarthy, Kodama and Williams. We use the plabic networks in the disk introduced by Postnikov to parametrize positroid cells in totally nonnegative Grassmannians. The boundary of the disk corresponds to the rational curve associated to the soliton data in the direct spectral problem, and the bicolored graph is the dual of a reducible curve G which is the rational degeneration of a regular Mcurve whose genus g equals the number of faces of the network diminished by one. We assign systems of edge vectors to the planar bicolored networks. The system of relations satisfied by them has maximal rank and may be reformulated in the form of edge signatures.. We prove that the components of the edge vectors are rational in the edge weights with subtraction free denominators and provide their explicit expressions in terms of conservative and edge flows. The edge vectors rule the value of the KP wave function at the double points of G, whereas the signatures at the vertices rule the position of the divisor in the ovals. We prove that the divisor satisfies the conditions of DubrovinNatanzon for real finitegap solutions: there is exactly one divisor point in each oval except for the one containing the essential singularity of the wave function. The divisor may be explicitly computed using the linear relations at the vertices of the network. We explain the role of moves and reductions in the transformation of both the curve and the divisor for given soliton data, and apply our construction to some examples.
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 arXiv:
 arXiv:1801.00208
 Bibcode:
 2018arXiv180100208A
 Keywords:

 Mathematical Physics;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 Version 1: 124 pages, 55 figures. The revised version of the first part of Version 1 (Lenetwork case) is now in arXiv:1805.05641. Version 2: 123 pages, 47 figures: it is the fully revised version of the second part of Version 1 (all plabic networks) and it contains several new results. Version 3: Section 6.2 (v2) has become section 7 and has been fully revised