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.
Waves on a fluid surface show a very complex behavior in general. Only under special circumstances can we expect to observe a more regular pattern. For shallow water waves, the Kadomtsev-Petviashvili (KP) equation (-4 u t + u xxx + 6 uu x ) x + 3 u yy = 0 (where e.g. u t = ∂u/∂t) provides an approximation under the conditions that the wave dominantly travels in the x-direction, the wave length is long as compared with the water depth, and the effect of the nonlinearity is about the same order as that of dispersion. 1 More precisely, this is the KP-II equation, but we will write KP, for short. It generalizes the famous Korteweg-deVries (KdV) equation, which describes waves moving in only one spatial dimension. Although the KdV equation is much better established as an approximation of the more general water wave equations, recent studies also confirm the physical relevance of KP [9].
In [2] we studied the soliton solutions of the KP equation in a tropical approximation, which reduces them to networks formed by line segments in the xy-plane, evolving in time t. A subclass corresponds to evolutions (in time t) in the set of (planar) rooted binary trees. At transition events, the binary tree type changes (through a tree that is not binary). It turned out that the time evolution is simply given by right rotation in a tree, and the solution evolves according to a maximal chain of a Tamari lattice [24,25], see Figure 1.
Figure 1: The Tamari lattice T 4 in terms of rooted binary trees. The top node shows a left comb tree that represents the structure of a certain KP line soliton family (with six asymptotic branches in the xy-plane) as t → -∞. A label ijkl assigned to an edge indicates the transition time t ijkl at which the soliton graph changes its tree type via a ‘rotation’ (see Section 3). The values of the parameters, on which the solutions depend, determine the linear order of the ‘critical times’ t ijkl , and thus decide which chain is realized. The bottom node shows a right comb tree, which represents the tree type of the soliton as t → ∞. The special family of solutions thus splits into classes corresponding to the maximal chains of T 4 . For each Tamari lattice, there is a family of KP line solitons that realizes its maximal chains in this way.
In this realization of Tamari lattices, the underlying set consists of states of a physical system, here the tree-types of a soliton configuration in the xy-plane. The Tamari poset (partially ordered set) structure describes the possible ways in which these states are allowed to evolve in time, starting from an initial state (the top node) and ending in a final state (the bottom node).
Figure 2 displays a solution evolving via a tree rotation, and further provides an idea how this can be understood in terms of an arrangement of planes in three-dimensional space-time (after idealizing line soliton branches to lines in the xy-plane, see Section 3).
In this work we show that the classification of possible evolutions of tree-shaped KP line solitons involves higher Bruhat orders [17][18][19]30]. Moreover, we are led to associate with each higher Bruhat order a higher Tamari order via a surjection, in a way different from what has been considered previously. There is some evidence that our higher Tamari orders coincide with ‘higher Stasheff-Tamari posets’ introduced by Kapranov and Voevodsky [8] (also see [3]), but a closer comparison will not be undertaken in this work.
In Section 2, we briefly describe the general class of KP soliton solutions. In Section 3, we concentrate on the abovementioned subclass of tree-shaped solutions, in the tropical approximation, and somewhat improve results in [2]. Section 4 recalls results about higher Bruhat orders and extracts from the analysis of tree-shaped KP line solitons a reduction to higher Tamari orders. Section 7 proposes a hierarchy of monoids To the right is a corresponding space-time view in terms of intersecting planes (here time flows upward).
that expresses the hierarchical structure present in the KP soliton problem. This makes contact with simplex equations [1,6,13,29] and provides us with an algebraic method to construct higher Bruhat and higher Tamari orders. Section 8 contains some additional remarks. Throughout this work, Hasse diagrams of posets will be displayed upside down (i.e. with the lowest element(s) at the top).
The line soliton solutions of the KP-II equation are parametrized by the totally non-negative Grassmannians Gr ≥ n,M +1 [9], which is easily recognized in the Wronskian form of the solutions. Translating the KP equation via u = 2 log(τ ) xx , into a bilinear equation in the variable τ , these solutions are given by
where
Here t (r) , r = 1, . . . , M , are independent real variables that may be regarded as coordinates on R M , and we set t (1) = x, t (2) = y, t (3) = t. The variables t (r) , r > 3, are the additional evolution variables that appear in the KP hierarchy, which extends the KP
This content is AI-processed based on open access ArXiv data.
