Translationally invariant nonlinear Schrodinger lattices

Abstract

Persistence of stationary and traveling single-humped localized solutions in the spatial discretizations of the nonlinear Schrodinger (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 and a necessary condition for existence of single-humped traveling solutions. Other constraints on the nonlinear function are found from the condition that the differential advance-delay equation for traveling solutions admits a reduction to an integrable normal form given by a third-order differential equation. This reduction also gives a necessary condition for existence of single-humped traveling solutions. The nonlinear function which admits both reductions defines a two-parameter family of discrete NLS equations which generalizes the integrable Ablowitz--Ladik lattice.

24 pages, 4 figures

References in corpus (1)

• Discrete Nonlinear Schrodinger Equations Free of the Peierls-Nabarro Potential

Cited by in corpus (7)

• Generalized Neighbor-Interaction Models Induced by Nonlinear Lattices
• Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential
• Comparative Study of Different Discretizations of the ${\varphi }^4$ Model
• High-speed kinks in a generalized discrete ${\varphi }^4$ model
• Exact static solutions to a translationally invariant discrete ${\varphi }^4$ model
• Stability, mobility and power currents in a two-dimensional model for waveguide arrays with nonlinear coupling
• Exact Moving and Stationary Solutions of a Generalized Discrete Nonlinear Schrodinger Equation