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
KP Solitons, Higher Bruhat and Tamari Orders
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 planarExpand
Matrix KP: tropical limit and Yang–Baxter maps
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 itsExpand
Kadomtsev-Petviashvili II equation: Structure of asymptotic soliton webs
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 websExpand
Matrix Kadomtsev–Petviashvili Equation: Tropical Limit, Yang–Baxter and Pentagon Maps
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 whoseExpand
Matrix Boussinesq solitons and their tropical limit
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", asExpand
Matrix KP: tropical limit, Yang-Baxter and pentagon maps
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 solitonExpand
Vertex dynamics in multi-soliton solutions of Kadomtsev–Petviashvili II equation
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 withExpand
KP solitons and total positivity for the Grassmannian
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 isExpand
Signatures of partition functions and their complexity reduction through the KP II equation
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 notionExpand
...
1
2
3
...

References

SHOWING 1-10 OF 60 REFERENCES
Soliton solutions of the Kadomtsev-Petviashvili II equation
We study a general class of line-soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation by investigating the Wronskian form of its tau-function. We show that, in addition to theExpand
Classification of the line-soliton solutions of KPII
In the previous papers (notably, Kodama Y 2004 J. Phys. A: Math. Gen. 37 11169–90, Biondini G and Chakravarty S 2006 J. Math. Phys. 47 033514), a large variety of line-soliton solutions of theExpand
On a family of solutions of the Kadomtsev–Petviashvili equation which also satisfy the Toda lattice hierarchy
We describe the interaction pattern in the x–y plane for a family of soliton solutions of the Kadomtsev–Petviashvili (KP) equation, The solutions considered also satisfy the finite Toda latticeExpand
KP solitons in shallow water
The main purpose of the paper is to provide a survey of our recent studies on soliton solutions of the Kadomtsev–Petviashvili (KP) equation. The KP equation describes weakly dispersive and smallExpand
On the construction of the KP line-solitons and their interactions
The line-soliton solutions of the Kadomtsev–Petviashvili (KP) equation are investigated in this article using the τ-function formalism. In particular, the Wronskian and the Grammian forms of theExpand
Elastic and inelastic line-soliton solutions of the Kadomtsev-Petviashvili II equation
TLDR
This work investigates a general class of multi-solitons which were not previously studied, and which do not in general conserve the number of line solitons after interaction. Expand
Line‐soliton solutions of the KP equation
A review of recent developments in the study and classification of the line‐soliton solutions of the Kadomtsev‐Petviashvili (KP) equation is provided. Such solution u(x, y, t) is defined by a pointExpand
Line soliton interactions of the Kadomtsev-Petviashvili equation.
  • G. Biondini
  • Physics, Medicine
  • Physical review letters
  • 2007
TLDR
It is shown that the (2N-1)!! types of solutions are uniquely characterized in terms of the individual soliton parameters, and the soliton position shifts arising from the interactions are calculated. Expand
On a family of solutions of the KP equation which also satisfy the Toda lattice hierarchy
We describe the interaction pattern in the x-y plane for a family of soliton solutions of the Kadomtsev-Petviashvili (KP) equation, (−4ut +uxxx+6uux)x +3uyy = 0. Those solutions also satisfy theExpand
FAST TRACK COMMUNICATION: Soliton solutions of the KP equation with V-shape initial waves
We consider the initial value problems of the Kadomtsev-Petviashvili (KP) equation for symmetric V-shape initial waves consisting of two semi-infinite line solitons with the same amplitude. NumericalExpand
...
1
2
3
4
5
...