# KP Solitons, Higher Bruhat and Tamari Orders

@article{Dimakis2012KPSH, title={KP Solitons, Higher Bruhat and Tamari Orders}, author={Aristophanes Dimakis and Folkert Mueller-Hoissen}, journal={arXiv: Combinatorics}, year={2012}, pages={391-423} }

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.

#### 13 Citations

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We briefly describe recent results about higher Bruhat and Tamari orders, the associated simplex equations and generalizations (polygon equations) of the pentagon equation, and the appearance of… Expand

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The set of triangulations of a cyclic polytope possesses two a priori different partial orders, known as the higher Stasheff–Tamari orders. The first of these orders was introduced by Kapranov and… Expand

Homotopy Type of Intervals of the Second Higher Bruhat Orders

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This work proves that every interval is either contractible or homotopy equivalent to a sphere in the higher Bruhat order, which partially proves a conjecture due to Reiner. Expand

Simplex and Polygon Equations

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It is shown that higher Bruhat orders admit a decomposition into a higher Tamari order, the corresponding dual Tamari order, and a \mixed order". We describe simplex equations (including the… Expand

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A functional of the solution of the Kadomtsev–Petviashvili II equation maps multi-soliton solutions onto systems of vertices—structures that are localized around soliton junctions. A solution with… Expand

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We prove two related conjectures concerning the higher Bruhat orders and the first higher Stasheff–Tamari orders. Namely, we prove the conjecture of Danilov, Karzanov, and Koshevoy that every… Expand

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We consider equations that formally resemble a matrix Riemann (or Hopf) equation in the framework of bidifferential calculus. With different choices of a first-order bidifferential calculus, we… Expand

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