On the \(3D\) consistency of a Grassmann extended lattice Boussinesq system. (English) Zbl 1479.37079
Summary: In this paper, we formulate a “Grassmann extension” scheme for constructing noncommutative (Grassmann) extensions of Yang-Baxter maps together with their associated systems of P\(\Delta\)Es, based on the ideas presented in [the author and T. E. Kouloukas, J. Math. Phys. 59, No. 6, 063506, 13 p. (2018; Zbl 1395.37046)]. Using this scheme, we first construct a Grassmann extension of a Yang-Baxter map which constitutes a lift of a lattice Boussinesq system. The Grassmann-extended Yang-Baxter map can be squeezed down to a novel, integrable, Grassmann lattice Boussinesq system, and we derive its \(3D\)-consistent limit. We show that some systems retain their \(3D\)-consistency property in their Grassmann extension.
MSC:
| 37K60 | Lattice dynamics; integrable lattice equations |
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 16T25 | Yang-Baxter equations |
| 39A36 | Integrable difference and lattice equations; integrability tests |
Citations:
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