Translationally invariant nonlinear Schrödinger lattices

The persistence of stationary and travelling single-humped localized solutions in the spatial discretizations of the nonlinear Schrödinger (NLS) equation is addressed. The discrete NLS equation with the most general cubic polynomial function is considered. Constraints on the nonlinear function are found from the condition that the second-order difference equation for stationary solutions can be reduced to the first-order difference map. The discrete NLS equation with such an exceptional nonlinear function is shown to have a conserved momentum but admits no standard Hamiltonian structure. It is proved that the reduction to the first-order difference map gives a sufficient condition for existence of translationally invariant single-humped stationary solutions. Another constraint on the nonlinear function is found from the condition that the differential advance–delay equation for travelling solutions admits a reduction to an integrable normal form given by a third-order differential equation. This reduction gives a necessary condition for existence of single-humped travelling solutions. The nonlinear function which admits both reductions defines a four-parameter family of discrete NLS equations which generalizes the integrable Ablowitz–Ladik lattice. Particular travelling solutions of this family of discrete NLS equations are written explicitly.