Validity of the NLS approximation for periodic quantum graphs

DOI10.1007/s00030-016-0417-7zbMath1367.35156OpenAlexW2556710322MaRDI QIDQ2397178

Dmitry E. Pelinovsky, Guido Schneider, Steffen Gilg

Publication date: 29 May 2017

Published in: NoDEA. Nonlinear Differential Equations and Applications (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s00030-016-0417-7

zbMATH Keywords

amplitude equationnonlinear Schrödinger (NLS) equationFloquet-Bloch spectrumBloch wave analysisperiodic quantum graph

Mathematics Subject Classification ID

Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs (35P30) NLS equations (nonlinear Schrödinger equations) (35Q55) Existence problems for PDEs: global existence, local existence, non-existence (35A01) Time-dependent Schrödinger equations and Dirac equations (35Q41) Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness (35A02) PDEs on graphs and networks (ramified or polygonal spaces) (35R02)

Related Items (17)

Standing waves on quantum graphs ⋮ Numerical simulations on nonlinear quantum graphs with the GraFiDi library ⋮ Nonlinear Schrödinger equations on periodic metric graphs ⋮ Nonlinear dynamics of modulated waves on graphene like quantum graphs ⋮ Breather solutions for a semilinear Klein-Gordon equation on a periodic metric graph ⋮ Nonlinear instability of half-solitons on star graphs ⋮ Ground States of the L 2-Critical NLS Equation with Localized Nonlinearity on a Tadpole Graph ⋮ Orbital instability of standing waves for NLS equation on star graphs ⋮ On the nonlinear Dirac equation on noncompact metric graphs ⋮ Bifurcations of standing localized waves on periodic graphs ⋮ Quantum graphs and dimensional crossover: the honeycomb ⋮ Construction of breather solutions for nonlinear Klein-Gordon equations on periodic metric graphs ⋮ Drift of Spectrally Stable Shifted States on Star Graphs ⋮ Edge-localized states on quantum graphs in the limit of large mass ⋮ An overview on the standing waves of nonlinear Schrödinger and Dirac equations on metric graphs with localized nonlinearity ⋮ Existence of the ground state for the NLS with potential on graphs ⋮ Mass-constrained ground states of the stationary NLSE on periodic metric graphs

Cites Work

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• Threshold phenomena and existence results for NLS ground states on metric graphs
• The NLS approximation makes wrong predictions for the water wave problem in case of small surface tension and spatially periodic boundary conditions
• Constrained energy minimization and orbital stability for the NLS equation on a star graph
• NLS ground states on graphs
• Bifurcations of standing localized waves on periodic graphs
• On the growth of high Sobolev norms of solutions for KdV and Schrödinger equations
• Variational properties and orbital stability of standing waves for NLS equation on a star graph
• Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential
• Bounds on the tight-binding approximation for the Gross-Pitaevskii equation with a periodic potential
• Justification of the nonlinear Schrödinger equation in spatially periodic media
• On the spectra of carbon nano-structures
• Justification of the lattice equation for a nonlinear elliptic problem with a periodic potential
• Semigroups of linear operators and applications to partial differential equations
• Stable transport of information near essentially unstable localized structures
• Schrödinger operators on zigzag nanotubes
• Nonlinear Schrödinger equation on graphs: recent results and open problems
• Survey on global existence in the nonlinear Dirac equations in one dimension
• Decisiveness of the spectral gaps of periodic Schrödinger operators on the dumbbell-like metric graph
• The validity of modulation equations for extended systems with cubic nonlinearities
• Validity and Limitation of the Newell‐Whitehead Equation
• On Tight‐Binding Approximations in Optical Lattices
• Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential

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