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 nonlinear map in the case of a more general matrix KP equation. We also consider the corresponding Korteweg–deVries reduction. Furthermore, exploiting the fine structure of soliton interactions in the tropical limit, we obtain an apparently 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.
Highlights
A line soliton solution of the scalar Kadomtsev–Petviashvili (KP-II) equation is, at fixed time t, an exponentially localized wave on a plane
From (2.2), we find that the pure soliton solutions of the pKPK equation are given by φ = F, τ with τ := eθ2 det, F := −eθ2 θ1 eθ(P1) adj( ) e−θ(Q1) χ1, (3.3) (3.4)
The tropical limit of a soliton solution at a fixed time t has support on the visible parts of the boundaries between the regions associated with phases appearing in τ
Summary
A line soliton solution of the scalar Kadomtsev–Petviashvili (KP-II) equation (see, e.g., [16]) is, at fixed time t, an exponentially localized wave on a plane. It is known [12,24] that this yields a Yang–Baxter map, i.e., a set-theoretical solution of the (quantum) Yang–Baxter equation ( see [1,23] for the case of the vector Nonlinear Schrödinger equation) Not surprisingly, this is a feature preserved in the tropical limit. In case of a vector KdV equation, i.e., KdVK with n = 1, it is given by an R-matrix, a linear map solution of the Yang–Baxter equation. The polarization values are related by a linear Yang–Baxter map, represented by an R-matrix, which does not depend on the independent variables x, y, t, but only on the “spectral parameters” of the soliton solution.
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