Constructions of Totally Non-Negative Pfaffian

TL;DR

This work addresses the problem of constructing totally non-negative Pfaffians (TNNP) for BKP-τ functions, ensuring that every principal Pfaffian of even size is nonnegative. It develops two complementary construction strategies: graph-theoretic methods based on planar perfect matchings and chord diagrams, and lattice-path approaches via Dyck paths, both yielding Pfaffian representations that guarantee nonnegativity. A Pfaffian factorization $A=LDL^T$ with a skew-symmetric tridiagonal $D$ and a totally nonnegative $L$ provides a concrete pathway to extend TNNP to larger matrix sizes, with explicit formulas connecting subpfaffians to Pfaffians of subgraphs or Dyck configurations. These results furnish a constructive framework for generating TNNP-associated BKP soliton τ-functions and suggest avenues for extending to modified BKP and Veselov–Novikov integrable systems, including potential applications to soliton resonance structures.

Abstract

The totally non-negative pfaffian (TNNP) is define for a skew-symmetric matrix such that all the sub-pfaffians are non-negative. It appears in the pfaffian structure of $τ$-function for the non-singular web solitons of the BKP equation . One constructs TNNP using the Perfect matching, chord diagram and the Dyck paths enumeration.Its tridiagonal form is also investigated.

Figures (6)

• Figure 1: This plane graph $G$ has six vertices, nine edges and $M(G)=4$. There are four boundary vertices.
• Figure 2: The plane graph $\textbf{G}$ at left-hand side has eight vertices with six boundary vertices $(n=3, a_1=1, a_2=2, a_3=3, \cdots, a_6=6)$, thirteen edges and $M( \textbf{G})=7$. Also, $M( \textbf{G} \backslash {1,2,3,4})=2$ (k=2) and the matrix $A$ is a TNNP constructed from the graph ${\textbf{G}}$.
• Figure 3: This plane graph at left-hand side has 14 vertices, 14 edges and $M(G)=1$. There are six numbered boundary vertices. The right-hand is the soliton solution (\ref{['bp6']}) of BKP from the plane graph with $p_1=2, p_2=1.5, p_3=1, p_4=0.6, p_5=0.3, p_6=0.1$ . The numbers in blue color are the sub-pfaffians with dominant exponentials in the $\tau$-function (\ref{['bp6']}) at different regions. The numbers in red color are the $[ij]$-solitons localized at the boundaries of two distinct regions where a balance exists between two dominant exponentials.
• Figure 4: The interior points are even number.
• Figure 5: Two Dyck paths are shown here with a=-1, b=2. There are $C(3)=5$ Dyck paths.

Theorems & Definitions (19)

• Definition 1.1
• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Definition 2.5
• Theorem 2.6
• proof
• Theorem 2.7
• Theorem 2.8