Tropical limit of line-soliton solutions of the KP-II equation

The Kadomtsev-Petviashvili (KP) equation

(-4 ut + uxxx + 6 u ux)x ± 3 uyy = 0

is an extension of the famous Korteweg-deVries (KdV) equation to two spatial dimensions. The sign of the term uyy distinguishes between two versions with qualitatively rather different behavior, KP-I ("-") and KP-II ("+").

Each solution of the KdV equation becomes a solution of KP if we consider it to be constant in the y-direction. A KdV soliton then becomes a "line soliton" solution of KP, parallel to the y-axis and moving in x-direction. Multi-solitons of KdV are parallel line solitons of KP. KP-II also allows oblique line solitons. Moreover, it has exact solutions forming network-like structures. Whereas in KP-II line solitons are stable, this is not the case for KP-I. But in contrast to KP-II, KP-I admits stable excitations that are (rationally) localized in both spatial dimensions, so-called lumps.

Tree-shaped KP-II line soliton solutions and Tamari lattices

There is subset of KP-II line soliton solutions given by

u = 2 (log τ)xx           τ = ∑k=1,...,n+1 eθk           θk = pk x + pk2 y + pk3 t + ck

with real parameters pk, ck, k=1,...,n+1. In the region where θj (for some j) dominates, we have

log τ = θj + log( 1 + ∑k ≠ j   e-(θjk) )

Therefore

log τ ≈ max(θ1,...,θn+1)


Tamari lattice T4

In this "tropical limit" (which can be made precise via Maslov dequantization), line solitons are boundaries between dominating phase regions, because u vanishes in the interior of any such region (since θj is linear in x). This translates line soliton solutions (for fixed parameters) into piecewise linear graphs, and their evolutions sweep out piecewise linear hypersurfaces in (2+1)-dimensional space-time.

For fixed parameters, and at an instant of time, any solution from the above class in this way determines a rooted binary tree in the xy-plane. This tree is binary except at special values of time, where a transition takes place between rooted binary tree types. Such an event is characterized as a coincidence of four different phases.

It turned out that any KP-II soliton solution from the above class corresponds to a maximal chain of a Tamari lattice. This correspondence involves a kind of discretization of the evolution, since we consider trees of the same "morphological type" as being equivalent. Hence the classification concentrates on determining the events at which the form of the tree changes. The corresponding analysis makes essential use of the higher evolution variables of the KP hierarchy (which extends the KP equation to an infinite family of "commuting" PDEs).

For each soliton number n+1 there is a Tamari lattice Tn. In the case under consideration, its vertices are represented by rooted binary trees with n+1 legs, and the order is given by right rotation in a tree. These lattices first appeared in 1951 in Dov Tamari's thesis at the Sorbonne in Paris, originating from a study of non-associative binary operations (the right rotation in a tree corresponds to a rightward application of the associativity law in a properly bracketed monomial). Tn forms a poset structure on a polytope in n-1 dimensions, called associahedron or Stasheff polytope. The figure shows T4 (as a Hasse diagram). Each of its maximal chains corresponds to a possible soliton evolution (time proceeds from top to bottom). A tuple of four numbers indicates a coincidence of the corresponding four phases (which characterizes a transition event).

The analysis also makes contact with so-called higher Bruhat orders (Y. Manin and V. Schechtman) and leads to higher Tamari orders, which generalize the above Tamari order. This work by Aristophanes Dimakis and Folkert Mueller-Hoissen started in 2010.