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This article is cited in 7 scientific papers (total in 7 papers)
Matrix Kadomtsev–Petviashvili equation: Tropical limit, Yang–Baxter and pentagon maps
A. Dimakisa, F. Müller-Hoissenb a Department of Financial and Management Engineering,
University of the Aegean, Chios, Greece
b Max-Planck-Institut für Dynamik und Selbstorganisation,
Göttingen, Germany
Abstract:
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 tropical limit graph has the form of a rooted and generically binary tree and also solutions whose limit graph comprises two relatively inverted such rooted tree graphs. The distribution of polarizations over the lines constituting the graph is fully determined by a parameter-dependent binary operation and a Yang–Baxter map (generally nonlinear), which becomes linear in the vector KP case and is hence given by an $R$-matrix. The parameter dependence of the binary operation leads to a solution of the pentagon equation, which has a certain relation to the Rogers dilogarithm via a solution of the hexagon equation, the next member in the family of polygon equations. A generalization of the $R$-matrix obtained in the vector KP case also solves a pentagon equation. A corresponding local version of the latter then leads to a new solution of the hexagon equation.
Keywords:
soliton, KP equation, Yang–Baxter map, pentagon equation, hexagon equation, tropical limit, binary tree, dilogarithm.
Received: 28.09.2017
Citation:
A. Dimakis, F. Müller-Hoissen, “Matrix Kadomtsev–Petviashvili equation: Tropical limit, Yang–Baxter and pentagon maps”, TMF, 196:2 (2018), 254–265; Theoret. and Math. Phys., 196:2 (2018), 1164–1173
Linking options:
https://www.mathnet.ru/eng/tmf9468https://doi.org/10.4213/tmf9468 https://www.mathnet.ru/eng/tmf/v196/i2/p254
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Abstract page: | 237 | Full-text PDF : | 40 | References: | 26 | First page: | 9 |
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