KP Solitons, Higher Bruhat and Tamari Orders

Abstract

In a tropical approximation, any tree-shaped line soliton solution, a member of the simplest class of soliton solutions of the Kadomtsev-Petviashvili (KP-II) equation, determines a chain of planar rooted binary trees, connected by right rotation. More precisely, it determines a maximal chain of a Tamari lattice. We show that an analysis of these solutions naturally involves higher Bruhat and higher Tamari orders.

Matrix KP: tropical limit and Yang–Baxter maps

“…From our previous work [2,3] about tree-shaped soliton solutions of the scalar KP equation, nontrivial solutions of the pentagon equation were expected to emerge in case of a matrix version of the KP equation. We confirmed this in the present work.…”

“…Contour plots of τ trop for fixed t in the xy-plane, with (A, M ) = (2, 2), (3, 1),(1,3),(3,2), respectively. The resulting graph (here with blue lines) divides the xy-plane into dominating phase regions U ai .Example 2.1.…”

Higher Bruhat and Tamari Orders and their Realizations

“…It realizes the Tamari order T(3, 2) and actually appears in nature as a "Miles resonance" on a fluid surface, mathematically described by a special soliton solution of the famous KP equation. Indeed, in a "tropical limit", where the above maximum function shows up, a subclass of KP soliton solutions realizes all the Tamari lattices T(N, N -3) in terms of rooted binary trees [4,5]. Time evolution is then given by right rotation in a tree, which translates to the rightward application of the associativity law in Tamari's original presentation of the lattices (in terms of bracketings of a monomial of fixed length [10]).…”

“…This includes a revision of the relation between higher Bruhat orders and simplex equations [2,3], a decomposition of higher Bruhat orders, the resulting Tamari orders (expected to be equivalent to higher Stasheff -Tamari orders), and a new family of equations associated with the latter. We finally recall the occurrence of higher Bruhat and Tamari orders in a "tropical limit" of solitons of the famous Kadomtsev-Petviashvili (KP) equation [4,5]. (1, 2,3) (2,1,3) (2,3,1) (3, 2,1)…”

Homotopy Type of Intervals of the Second Higher Bruhat Orders

“…Introduced by Manin and Schechtman, the higher Bruhat orders have many equivalent interpretations, including single-element extensions of an alternating matroid, cubical tilings of a cyclic zonotope, and "admissible" permutations of d-subsets of [n] up to a suitable equivalence; see [17,Theorem 4.1], [9], or [8]. The higher Bruhat orders have appeared in a wide variety of contexts, including higher categories and Zamolodchikov's tetrahedral equation [8], soliton solutions of the Kadomtsev-Petviashvili equation [5], and the multidimensional cube recurrence [7].…”