Lie symmetries and their local determinacy for a class of differential-difference equations

Differential-difference equations (DDEs) u(n)((k))(t) = F-n(t, u(n+a),..., u(n+b)) for k greater than or equal to 2 are studied for their differential Lie symmetries. We observe that while nonintrinsic Lie symmetries do exist in such DDEs, a great many admit only the intrinsic ones. We also propose a mechanism for automating symmetry calculations for fairly general DDEs, with a variety of features exemplified. In particular, the Fermi-Pasta-Ulam system is studied in detail and its new similarity solutions given explicitly. (C) 1998 Published by Elsevier Science B.V.