Discrete Nonlinear Schrodinger Equations Free of the Peierls-Nabarro Potential

Abstract

We derive a class of discrete nonlinear Schr{ö}dinger (DNLS) equations for general polynomial nonlinearity whose stationary solutions can be found from a reduced two-point algebraic problem. It is demonstrated that the derived class of discretizations contains subclasses conserving classical norm or a modified norm and classical momentum. These equations are interesting from the physical standpoint since they support stationary discrete solitons free of the Peierls-Nabarro potential. As a consequence, even in highly-discrete regimes, solitons are not trapped by the lattice and they can be accelerated by even weak external fields. Focusing on the cubic nonlinearity we then consider a small perturbation around stationary soliton solutions and, solving corresponding eigenvalue problem, we (i) demonstrate that solitons are stable; (ii) show that they have two additional zero-frequency modes responsible for their effective translational invariance; (iii) derive semi-analytical solutions for discrete solitons moving at slow speed. To highlight the unusual properties of solitons in the new discrete models we compare them with that of the classical DNLS equation giving several numerical examples.

arXiv commentMisprints noticed in the journal publication are corrected [in Eq. (1) and Eq. (34)]

References in corpus (15)

• Discrete Solitons and Breathers with Dilute Bose-Einstein Condensates
• Bright gap solitons of atoms with repulsive interaction
• Dispersion management for atomic matter waves
• Kink dynamics in a novel discrete sine-Gordon system
• Soliton mobility in nonlocal optical lattices
• Topological discrete kinks
• Translationally invariant discrete kinks from one-dimensional maps
• Exact Solutions of the Saturable Discrete Nonlinear Schrodinger Equation
• Travelling kinks in discrete phi^4 models
• Discrete Klein-Gordon models with static kinks free of the Peierls-Nabarro potential
• Exact solitary wave solutions for a discrete $\lambda {\varphi }^4$ field theory in 1+1 dimensions
• Exact static solutions for discrete ${\varphi }^4$ models free of the Peierls-Nabarro barrier: Discretized first integral approach
• Standard Nearest Neighbor Discretizations of Klein-Gordon Models Cannot Preserve Both Energy and Linear Momentum
• Kinks in dipole chains
• On a Class of Spatial Discretizations of Equations of the Nonlinear Schrodinger Type

Cited by in corpus (12)

• Translationally invariant nonlinear Schrodinger lattices
• Generalized Neighbor-Interaction Models Induced by Nonlinear Lattices
• The inverse problem for the Gross - Pitaevskii equation
• Exact static solutions for discrete ${\varphi }^4$ models free of the Peierls-Nabarro barrier: Discretized first integral approach
• Standing lattice solitons in the discrete NLS equation with saturation
• Comparative Study of Different Discretizations of the ${\varphi }^4$ Model
• High-speed kinks in a generalized discrete ${\varphi }^4$ model
• Enhanced mobility of quantum droplets in periodic lattices
• 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
• On a Class of Spatial Discretizations of Equations of the Nonlinear Schrodinger Type
• Exact Moving and Stationary Solutions of a Generalized Discrete Nonlinear Schrodinger Equation