[Paper Review] KP theory, plabic networks in the disk and rational degenerations of M--curves

Simonetta Abenda, P. G. Grinevich|arXiv (Cornell University)|Dec 30, 2017

Nonlinear Waves and Solitons60 references4 citations

TL;DR

This paper establishes a combinatorial and geometric bridge between totally non-negative Grassmannians and real finite-gap solutions of the KP equation via rational degenerations of M-curves. Using planar bicolored trivalent networks (plabic networks) in the disk, it constructs KP wave functions and divisors on degenerate curves, proving that divisor points lie exactly one per finite oval—confirming Dubrovin-Natanzon conditions—using edge vectors and signatures derived from conservative and edge flows with rational, subtraction-free expressions in edge weights.

ABSTRACT

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 degenerations of M-curves for the class of real regular multi-line 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 M-curve 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 Dubrovin-Natanzon for real finite-gap 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.

Motivation & Objective

• To extend the correspondence between totally non-negative Grassmannians and real regular divisors on rational degenerations of M-curves in the context of KP finite-gap theory.
• To provide a geometric and combinatorial realization of KP multi-line soliton solutions using planar bicolored trivalent networks in the disk.
• To prove that the KP divisor on the rational degeneration of an M-curve satisfies the Dubrovin-Natanzon reality conditions: exactly one point per finite oval and none in the oval with the essential singularity.
• To derive explicit, rational, subtraction-free expressions for edge vectors in terms of conservative and edge flows, enabling computation of the wave function at double points.
• To characterize the effect of Postnikov moves and reductions on the curve, divisor, and edge vectors, ensuring invariance of the divisor structure under equivalence transformations.

Proposed method

• Constructs a reducible rational curve Γ as the dual of a planar bicolored trivalent network (plabic network) in the disk, with genus g equal to the number of faces minus one.
• Assigns edge vectors to edges of the network, which encode the value of the KP wave function at double points of Γ.
• Derives rational expressions for edge vector components using conservative and edge flows, with denominators free of subtractions, based on Lam’s edge signature formalism.
• Uses vertex signatures and edge loop-erased walks to determine divisor point positions in the ovals of the M-curve, ensuring one point per finite oval.
• Applies gauge freedom and orientation choices to show invariance of the wave function and divisor structure under network transformations.
• Introduces moves (M1–M3) and reductions (R1–R3) to relate different networks representing the same soliton data, preserving the divisor and curve structure.

Experimental results

Research questions

• RQ1 How can totally non-negative Grassmannians be geometrically realized via rational degenerations of M-curves in the context of KP soliton theory?
• RQ2 What is the precise combinatorial and algebraic structure of the KP wave function and divisor on a rational degeneration of an M-curve?
• RQ3 How do edge vectors and their rational expressions in edge weights relate to the wave function at double points of the degenerate curve?
• RQ4 How do Postnikov moves and reductions affect the curve, divisor, and edge vectors while preserving the soliton solution?
• RQ5 Can the divisor on the degenerate curve be explicitly computed and shown to satisfy the Dubrovin-Natanzon reality conditions?

Key findings

• The components of the edge vectors are rational functions of the edge weights with subtraction-free denominators, explicitly expressed via conservative and edge flows.
• The wave function at double points of the degenerate curve is fully determined by the system of edge vectors derived from the network.
• The divisor points lie exactly one per finite oval of the M-curve, and none in the oval containing the essential singularity—confirming the Dubrovin-Natanzon reality condition.
• The position of divisor points is combinatorially determined by vertex signatures and edge loop-erased walks, ensuring consistency across gauge and orientation choices.
• Moves and reductions (M1–M3, R1–R3) preserve the divisor and curve structure, providing a complete equivalence relation on networks for a given soliton data.
• The construction provides a global parametrization of positroid cells via KP divisors, demonstrated explicitly in the case of Gr^{TP}(1,3).