# KP line solitons and Tamari lattices

@article{Dimakis2010KPLS, title={KP line solitons and Tamari lattices}, author={Aristophanes Dimakis and Folkert Mueller-Hoissen}, journal={arXiv: Mathematical Physics}, year={2010} }

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… Expand

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#### 30 Citations

KP Solitons, Higher Bruhat and Tamari Orders

- Mathematics, Physics
- 2012

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… Expand

Matrix KP: tropical limit and Yang–Baxter maps

- Physics, Mathematics
- 2017

We study soliton solutions of matrix Kadomtsev–Petviashvili (KP) equations in a tropical limit, in which their support at fixed time is a planar graph and polarizations are attached to its… Expand

Kadomtsev-Petviashvili II equation: Structure of asymptotic soliton webs

- Physics, Mathematics
- 2014

A wealth of observations, recently supported by rigorous analysis, indicate that, asymptotically in time, most multi-soliton solutions of the Kadomtsev-Petviashvili II equation self-organize in webs… Expand

Matrix Kadomtsev–Petviashvili Equation: Tropical Limit, Yang–Baxter and Pentagon Maps

- Mathematics
- 2018

In the tropical limit of matrix KP-II solitons, their support at a fixed time is a planar graph with “polarizations” attached to its linear parts. We explore a subclass of soliton solutions whose… Expand

Matrix Boussinesq solitons and their tropical limit

- Physics, Mathematics
- Physica Scripta
- 2019

We study soliton solutions of matrix "good" Boussinesq equations, generated via a binary Darboux transformation. Essential features of these solutions are revealed via their "tropical limit", as… Expand

Matrix KP: tropical limit, Yang-Baxter and pentagon maps

- Mathematics, Physics
- 2017

In the tropical limit of matrix KP-II solitons, their support at fixed time is a planar graph with "polarizations" attached to its linear parts. In this work we explore a subclass of soliton… Expand

Vertex dynamics in multi-soliton solutions of Kadomtsev–Petviashvili II equation

- Physics
- 2014

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

KP solitons and total positivity for the Grassmannian

- Mathematics, Physics
- 2011

Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili proposed a two-dimensional nonlinear dispersive wave equation now known as the KP equation. It is… Expand

Signatures of partition functions and their complexity reduction through the KP II equation

- Physics, Mathematics
- 2017

A statistical amoeba arises from a real-valued partition function when the positivity condition for pre-exponential terms is relaxed, and families of signatures are taken into account. This notion… Expand

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