Birational solutions to the set-theoretical 4-simplex equation. (English) Zbl 1518.39007
Summary: The 4-simplex equation is a higher-dimensional analogue of Zamolodchikov’s tetrahedron equation and the Yang-Baxter equation which are two of the most fundamental equations of mathematical physics. In this paper, we introduce a method for constructing 4-simplex maps, namely solutions to the set-theoretical 4-simplex equation, using Lax matrix refactorisation problems. Employing this method, we construct 4-simplex maps which at a certain limit give tetrahedron maps classified by R. M. Kashaev et al. [Theor. Math. Phys. 117, No. 3, 1402–1413 (1998; Zbl 0941.82018); translation from Teor. Mat. Fiz. 117, No. 3, 370–384 (1998)]. Moreover, we construct a Kadomtsev-Petviashvili type of 4-simplex map. Finally, we introduce a method for constructing 4-simplex maps which can be restricted on level sets to parametric 4-simplex maps using Darboux transformations of integrable PDEs. We construct a nonlinear Schrödinger type parametric 4-simplex map which is the first parametric 4-simplex map in the literature.
MSC:
| 39A36 | Integrable difference and lattice equations; integrability tests |
| 16T25 | Yang-Baxter equations |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 37K35 | Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems |
Keywords:
4-simplex equation; parametric 4-simplex maps; Zamolodchikov’s tetrahedron equation; parametric tetrahedron maps; Darboux transformations; NLS type equationsCitations:
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