Lie group analysis of a generalized Krichever-Novikov differential-difference equation

Abstract

The symmetry algebra of the differential-difference equation \documentclass[12pt]{minimal}\begin{document}$\dot{u}_n = N/D$\end{document}u̇n=N/D with D = un+1 − un−1 and N = P(un)un+1un−1 + Q(un)(un+1 + un−1) + R(un), where P, Q, and R are arbitrary analytic functions is shown to have the dimension 1 ⩽ dimL ⩽ 5. When P, Q, and R are specific second order polynomials in un (depending on 6 constants) this is the integrable discretization of the Krichever–Novikov equation. We find 3 cases when the arbitrary functions are not polynomials and the symmetry algebra satisfies dimL = 2. These cases are shown not to be integrable. The symmetry algebras are used to reduce the equations to purely difference ones. The symmetry group is also used to impose periodicity un+N = un and thus to reduce the differential-difference equation to a system of N coupled ordinary three points difference equations.