scholarly journals Geometric Monodromy around the Tropical Limit

2016 ◽  
Author(s):  
Yuto Yamamoto ◽  
Keyword(s):  
Tropical Limit

scholarly journals Tropical limit in statistical physics

2015 ◽  
Vol 379 (24-25) ◽  
pp. 1497-1502 ◽  
Author(s):  
M. Angelelli ◽  
B. Konopelchenko
Keyword(s):  
Statistical Physics ◽  
Tropical Limit

scholarly journals Matrix Boussinesq solitons and their tropical limit

2019 ◽  
Vol 94 (3) ◽  
pp. 035206 ◽  
Author(s):  
Aristophanes Dimakis ◽  
Folkert Müller-Hoissen ◽  
Xiao-Min Chen
Keyword(s):  
Tropical Limit

scholarly journals Matrix KP: tropical limit and Yang–Baxter maps

2018 ◽  
Vol 109 (4) ◽  
pp. 799-827 ◽  
Author(s):  
Aristophanes Dimakis ◽  
Folkert Müller-Hoissen
Keyword(s):  
Tropical Limit

Geometry in the tropical limit

2011 ◽  
Vol 59 (1) ◽  
pp. 57-73 ◽  
Author(s):  
I. Itenberg ◽  
G. Mikhalkin
Keyword(s):  
Tropical Limit

scholarly journals Tropical limit of log-inflection points for planar curves

2018 ◽  
Vol 209 (9) ◽  
pp. 1273-1286
Author(s):  
G. B. Mikhalkin ◽  
A. Renaudineau
Keyword(s):  
Planar Curves ◽  
Inflection Points ◽  
Tropical Limit

scholarly journals Tropical limit of matrix solitons and entwining Yang–Baxter maps

2020 ◽  
Vol 110 (11) ◽  
pp. 3015-3051
Author(s):  
Aristophanes Dimakis ◽  
Folkert Müller-Hoissen
Keyword(s):  
Soliton Solution ◽  
Toda Lattice ◽  
Soliton Solutions ◽  
Baxter Equation ◽  
Kp Equation ◽  
Lattice Equation ◽  
Soliton Interactions ◽  
The Matrix ◽  
Tropical Limit ◽  
Toda Lattice Equation

Abstract We consider a matrix refactorization problem, i.e., a “Lax representation,” for the Yang–Baxter map that originated as the map of polarizations from the “pure” 2-soliton solution of a matrix KP equation. Using the Lax matrix and its inverse, a related refactorization problem determines another map, which is not a solution of the Yang–Baxter equation, but satisfies a mixed version of the Yang–Baxter equation together with the Yang–Baxter map. Such maps have been called “entwining Yang–Baxter maps” in recent work. In fact, the map of polarizations obtained from a pure 2-soliton solution of a matrix KP equation, and already for the matrix KdV reduction, is not in general a Yang–Baxter map, but it is described by one of the two maps or their inverses. We clarify why the weaker version of the Yang–Baxter equation holds, by exploring the pure 3-soliton solution in the “tropical limit,” where the 3-soliton interaction decomposes into 2-soliton interactions. Here, this is elaborated for pure soliton solutions, generated via a binary Darboux transformation, of matrix generalizations of the two-dimensional Toda lattice equation, where we meet the same entwining Yang–Baxter maps as in the KP case, indicating a kind of universality.

scholarly journals The U ( n ) Gelfand–Zeitlin system as a tropical limit of Ginzburg–Weinstein diffeomorphisms

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170428
Author(s):  
A. Alekseev ◽  
J. Lane ◽  
Y. Li
Keyword(s):  
Integrable System ◽  
Open Dense Subset ◽  
Triangular Matrices ◽  
Natural Action ◽  
Finite Dimensional ◽  
Totally Positive ◽  
Positive Matrices ◽  
Lagrangian Tori ◽  
Tropical Limit ◽  
Funct Anal

In this paper, we show that the Ginzburg–Weinstein diffeomorphism of Alekseev & Meinrenken (Alekseev, Meinrenken 2007 J. Differential Geom. 76 , 1–34. (10.4310/jdg/1180135664)) admits a scaling tropical limit on an open dense subset of . The target of the limit map is a product , where is the interior of a cone, T is a torus, and carries an integrable system with natural action-angle coordinates. The pull-back of these coordinates to recovers the Gelfand–Zeitlin integrable system of Guillemin & Sternberg (Guillemin, Sternberg 1983 J. Funct. Anal. 52 , 106–128. (10.1016/0022-1236(83)90092-7)). As a by-product of our proof, we show that the Lagrangian tori of the Flaschka–Ratiu integrable system on the set of upper triangular matrices meet the set of totally positive matrices for sufficiently large action coordinates. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

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