Justification of the lattice equation for a nonlinear elliptic problem with a periodic potential

DOI10.1007/s00220-008-0640-0zbMath1162.47059arXiv0704.2121OpenAlexW3104445341MaRDI QIDQ1006307

Guido Schneider, Dmitry E. Pelinovsky, Robert S. MacKay

Publication date: 20 March 2009

Published in: Communications in Mathematical Physics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/0704.2121

zbMATH Keywords

nonlinear elliptic problemlattice equationBose-Einstein equationFourier Bloch decompositionGross-Pitaevskij equationGross-Pitaevsky equationShannon decomposition

Mathematics Subject Classification ID

Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics (81Q05) Nonlinear elliptic equations (35J60) Applications of operator theory in the physical sciences (47N50) Schrödinger operator, Schrödinger equation (35J10) Applications of operator theory to differential and integral equations (47N20) Additive difference equations (39A10) Linear difference operators (47B39)

Related Items (16)

Nonlinear Schrödinger equations with a multiple-well potential and a Stark-type perturbation ⋮ Bounds on the tight-binding approximation for the Gross-Pitaevskii equation with a periodic potential ⋮ Nonlinear Stark--Wannier Equation ⋮ Onsite and offsite bound states of the discrete nonlinear Schrödinger equation and the Peierls–Nabarro barrier ⋮ Validity of the NLS approximation for periodic quantum graphs ⋮ Unified Orbital Description of the Envelope Dynamics in Two-Dimensional Simple Periodic Lattices ⋮ Discrete diffraction managed solitons: Threshold phenomena and rapid decay for general nonlinearities ⋮ Breather solutions in periodic media ⋮ On Tight‐Binding Approximations in Optical Lattices ⋮ Discrete solitary waves in systems with nonlocal interactions and the Peierls-Nabarro barrier ⋮ Bifurcations of standing localized waves on periodic graphs ⋮ Bifurcation of gap solitons in periodic potentials with a periodic sign-varying nonlinearity coefficient ⋮ Construction of real-valued localized composite Wannier functions for insulators ⋮ Derivation of the tight-binding approximation for time-dependent nonlinear Schrödinger equations ⋮ Stationary states for nonlinear Schrödinger equations with periodic potentials ⋮ Nonlinear optical waveguide lattices: asymptotic analysis, solitons, and topological insulators

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