Preliminary group classification of the nonlinear differential-difference equations (Q1933632)

Combining Lie's infinitesimal method and the theory of the classification of abstract finite-dimensional Lie algebras, the authors give a preliminary group classification of the nonlinear differential-difference equation \[ \Delta_n\equiv\ddot{u}_n(t)-F_n(t, u_{n-1}(t), u_n(t), u_{n+1}(t))=0. \] It is proved that there exists only one equation admitting three-dimensional simple Lie algebras, and there exist 3, 11, 22, 13 inequivalent equations admitting one-, two-, three- and four-dimensional solvable Lie algebras, respectively. Also, there exist two inequivalent equations admitting five- and seven-dimensional indecomposable algebras of semi-direct sums of semi-simple and solvable Lie algebras. It should be mentioned, that this classification is incomplete and that it is not possible to give a complete group classification within the presented framework.