zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Excitation thresholds for nonlinear localized modes on lattices. (English) Zbl 0984.35147
Summary: We consider spatially localized and time-periodic solutions to discrete extended Hamiltonian dynamical systems (coupled systems of infinitely many ‘oscillators’ which conserve total energy). These play a central role as carriers of energy in models of a variety of physical phenomena. Such phenomena include nonlinear waves in crystals, biological molecules and arrays of coupled optical waveguides. In this paper we study excitation thresholds for (nonlinearly dynamically stable) ground state localized modes, sometimes referred to as ‘breathers’, for networks of coupled nonlinear oscillators and wave equations of nonlinear Schrödinger (NLS) type. Excitation thresholds are rigorously characterized by variational methods. The excitation threshold is related to the optimal (best) constant in a class of discrete interpolation inequalities related to the Hamiltonian energy. We establish a precise connection among d, the dimensionality of the lattice, 2σ+1, the degree of the nonlinearity, and the existence of an excitation threshold for discrete nonlinear Schrödinger systems (DNLS). We prove that if σ2/d, then ground-state standing waves exist if and only if the total power is larger than some strictly positive threshold, ν thresh (σ,d). This proves a conjecture of S. Flach, K. Kladko and R. S. MacKay [Phys. Rev. Lett. 78, No. 7, 1207-1210 (1997)] in the context of DNLS. We also discuss upper and lower bounds for excitation thresholds for ground states of coupled systems of NLS equations, which arise in the modelling of pulse propagation in coupled arrays of optical fibres.

MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
78A60Lasers, masers, optical bistability, nonlinear optics