KP line solitons and Tamari lattices

Abstract

The KP-II equation possesses a class of line soliton solutions which can be qualitatively described via a tropical approximation as a chain of rooted binary trees, except at "critical" events where a transition to a different rooted binary tree takes place. We prove that these correspond to maximal chains in Tamari lattices (which are poset structures on associahedra). We further derive results that allow to compute details of the evolution, including the critical events. Moreover, we present some insights into the structure of the more general line soliton solutions. All this yields a characterization of possible evolutions of line soliton patterns on a shallow fluid surface (provided that the KP-II approximation applies).

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.…”

“…The time evolution of a rooted tree-shaped KP soliton solution determines a sequence of tropical limit graphs, which are rooted binary trees, connected by right rotation in trees [2,3]. At which vertex such a right rotation takes place next, depends on the values of higher KP hierarchy evolution variables [2,3]. In case of a 5-soliton solution, we only need the next higher KP hierarchy variable.…”

“…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)…”