Symmetries of differential-difference dynamical systems in a two-dimensional lattice. (English) Zbl 1184.37044
From the abstract:
“The classification of differential-difference equations of the form \(\ddot{u}_{nm}=F_{nm} (t, \{u_{pq}\}|_{(p,q)\in \Gamma}) \) is considered according to their Lie point symmetry groups. The set \(\Gamma\) represents the point \((n,m)\) and its six nearest neighbors in a two-dimensional triangular lattice. It is shown that the symmetry group can be at most 12 dimensional for Abelian symmetry algebras and at most 13 dimensional for nonsolvable symmetry algebras.”
The aim is thus to classify the interaction functions \(F_{nm}\), which are assumed nonlinear and isotropic, according to the Lie point symmetries.
“The classification of differential-difference equations of the form \(\ddot{u}_{nm}=F_{nm} (t, \{u_{pq}\}|_{(p,q)\in \Gamma}) \) is considered according to their Lie point symmetry groups. The set \(\Gamma\) represents the point \((n,m)\) and its six nearest neighbors in a two-dimensional triangular lattice. It is shown that the symmetry group can be at most 12 dimensional for Abelian symmetry algebras and at most 13 dimensional for nonsolvable symmetry algebras.”
The aim is thus to classify the interaction functions \(F_{nm}\), which are assumed nonlinear and isotropic, according to the Lie point symmetries.
Reviewer: Jens Rademacher (Amsterdam)
MSC:
| 37J15 | Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010) |
| 70G65 | Symmetries, Lie group and Lie algebra methods for problems in mechanics |
| 20C35 | Applications of group representations to physics and other areas of science |
