Geodesic
Matrix Kadomtsev–Petviashvili equation: Tropical limit, Yang–Baxter and pentagon maps
Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 2, pp. 254-265 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Mots-clés : soliton, KP equation, hexagon equation
Keywords: Yang–Baxter map, pentagon equation, tropical limit, binary tree, dilogarithm. 
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     author = {A. Dimakis and F. M\"uller-Hoissen},
     title = {Matrix {Kadomtsev{\textendash}Petviashvili} equation: {Tropical} limit, {Yang{\textendash}Baxter} and pentagon maps},
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A. Dimakis; F. Müller-Hoissen. Matrix Kadomtsev–Petviashvili equation: Tropical limit, Yang–Baxter and pentagon maps. Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 2, pp. 254-265. http://geodesic.mathdoc.fr/item/TMF_2018_196_2_a4/

[1] A. Dimakis, F. Müller-Hoissen, Matrix KP: tropical limit and Yang–Baxter maps, arXiv: 1708.05694

[2] A. Dimakis, F. Müller-Hoissen, “KP line solitons and Tamari lattices”, J. Phys. A: Math. Theor., 44:2 (2011), 025203, 49 pp. | DOI | MR | Zbl

[3] A. Dimakis, F. Müller-Hoissen, “KP solitons, higher Bruhat and Tamari orders”, Associahedra, Tamari Lattices and Related Structures, Progress in Mathematics, 299, eds. F. Müller-Hoissen, J. Pallo, J. Stasheff, Birkhäuser, Basel, 2012, 391–423 | DOI | MR

[4] A. Dimakis, F. Müller-Hoissen, “KdV soliton interactions: a tropical view”, J. Phys.: Conf. Ser., 482:1 (2014), 012010, 11 pp. | DOI

[5] A. Dimakis, F. Müller-Hoissen, “Simplex and polygon equations”, SIGMA, 11 (2015), 042, 49 pp., arXiv: 1409.7855 | DOI | MR | Zbl

[6] M. M. Kapranov, V. A. Voevodsky, “2-Categories and Zamolodchikov tetrahedron equations”, Algebraic Groups and Their Generalizations: Quantum and Infinite-Dimensional Methods (Pennsylvania State University, University Park, PA, USA, July 6–26, 1991), Proceedings of Symposia in Pure Mathematics, 56, eds. W. J. Haboush, B. J. Parshall, AMS, Providence, RI, 1994, 177–259 | DOI | MR | Zbl

[7] R. Street, “Fusion operators and cocycloids in monoidal categories”, Appl. Categ. Struct., 6:2 (1998), 177–191 | DOI | MR | Zbl

[8] I. G. Korepanov, “Two-cocycles give a full nonlinear parameterization of the simplest 3–3 relation”, Lett. Math. Phys., 104:10 (2014), 1235–1261 | DOI | MR | Zbl

[9] R. M. Kashaev, “A simple model of $4$D-TQFT”, arXiv: 1405.5763

[10] R. Kashaev, “On realizations of Pachner moves in $4$d”, J. Knot Theory Ramifications, 24:13 (2015), 1541002, 13 pp. | DOI | MR | Zbl

[11] I. G. Korepanov, N. M. Sadykov, “Hexagon cohomologies and a cubic TQFT action”, arXiv: 1707.02847

[12] R. M. Kashaev, S. M. Sergeev, “On pentagon, ten-term, and tetrahedron relations”, Commun. Math. Phys., 195:2 (1998), 309–319 | DOI | MR | Zbl

[13] R. M. Kashaev, “O matrichnykh obobscheniyakh dilogarifma”, TMF, 118:3 (1999), 398–404 | DOI | DOI | MR | Zbl

[14] L. J. Rogers, “On function sum theorems connected with the series $\sum_{n=1}^\infty \frac{x^n}{n^2}$”, Proc. London Math. Soc. (2), s2-4:1 (1907), 169–189 | DOI | MR