The Discrete Nonlinear Schrödinger Equation and its Lie Symmetry Reductions

The Lie algebra L(h) of symmetries of a discrete analogue of the non-linear Schrodinger equation (NLS) is studied. A five-dimensional subspace of L(h), generated by both point and generalized symmetries, transforms into the five-dimensional point symme- try algebra L(0) of the NLS equation. We use the lowest symmetries to do symmetry reduction of the equation, thus obtaining explicit solutions and discrete analogues of elliptic functions.