On a Class of Spatial Discretizations of Equations of the Nonlinear Schrodinger Type

Abstract

We demonstrate the systematic derivation of a class of discretizations of nonlinear Schr{ö}dinger (NLS) equations for general polynomial nonlinearity whose stationary solutions can be found from a reduced two-point algebraic condition. We then focus on the cubic problem and illustrate how our class of models compares with the well-known discretizations such as the standard discrete NLS equation, or the integrable variant thereof. We also discuss the conservation laws of the derived generalizations of the cubic case, such as the lattice momentum or mass and the connection with their corresponding continuum siblings.

Submitted for publication in a journal on October 14, 2005

References in corpus (13)

• Theory of Nonlinear Matter Waves in Optical Lattices
• Pattern Forming Dynamical Instabilities of Bose-Einstein Condensates: A Short Review
• Translationally invariant discrete kinks from one-dimensional maps
• Vortices in Bose-Einstein Condensates: Some Recent Developments
• Travelling kinks in discrete phi^4 models
• Translationally invariant nonlinear Schrodinger lattices
• Discrete Klein-Gordon models with static kinks free of the Peierls-Nabarro potential
• Mobile Localization in nonlinear Schrodinger lattices
• Exact static solutions for discrete ${\varphi }^4$ models free of the Peierls-Nabarro barrier: Discretized first integral approach
• Nonintegrable Schrodinger Discrete Breathers
• Standard Nearest Neighbor Discretizations of Klein-Gordon Models Cannot Preserve Both Energy and Linear Momentum
• Discrete Nonlinear Schrodinger Equations Free of the Peierls-Nabarro Potential
• Kinks in dipole chains

Cited by in corpus (7)

• Generalized Neighbor-Interaction Models Induced by Nonlinear Lattices
• Discrete Nonlinear Schrodinger Equations Free of the Peierls-Nabarro 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