Matrix KP: tropical limit and Yang-Baxter maps

Abstract

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 constituting lines. There is a subclass of "pure line soliton solutions" for which we find that, in this limit, the distribution of polarizations is fully determined by a Yang-Baxter map. For a vector KP equation, this map is given by an R-matrix, whereas it is a non-linear map in case of a more general matrix KP equation. We also consider the corresponding Korteweg-deVries (KdV) reduction. Furthermore, exploiting the fine structure of soliton interactions in the tropical limit, we obtain a new solution of the tetrahedron (or Zamolodchikov) equation. Moreover, a solution of the functional tetrahedron equation arises from the parameter-dependence of the vector KP R-matrix.

arXiv comment23 pages, 9 figures, second version: some minor amendments, reformulations in Section 4, additional references [10] and [18]

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• Yang-Baxter maps and integrable dynamics
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Cited by in corpus (10)

• On non-abelian quadrirational Yang-Baxter maps
• Yang--Baxter maps, Darboux transformations, and linear approximations of refactorisation problems
• Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation
• Grassmannian-parameterized solutions to direct-sum polygon and simplex equations
• Birational solutions to the set-theoretical 4-simplex equation
• Tetrahedron maps, Yang-Baxter maps, and partial linearisations
• Two-component Yang-Baxter maps and star-triangle relations
• Second-order Symmetric Non-negative Latent Factor Analysis
• Tropical limit of matrix solitons and entwining Yang-Baxter maps
• Matrix Boussinesq solitons and their tropical limit