zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Traveling waves of bistable dynamics on a lattice. (English) Zbl 1050.37041
Summary: We prove the existence of stationary or traveling waves in a lattice dynamical system arising in the theory of binary phase transitions. The system allows infinite-range couplings with positive and negative weights. The allowance for negative coupling coefficients precludes the possibility of a maximum principle. Instead, a weakened type of ellipticity is stipulated that is used with spectral theory in a perturbative fixed point argument to construct a traveling wave when the nonlinearity is unbalanced and the coupling is sufficiently strong. When the nonlinearity is balanced, a variational technique is used to obtain stationary waves, which are then analyzed in more detail for strong couplings. From a physical perspective these models are important since long-range and indefinite interactions occur in nature and can lead to pattern formation. Our results provide conditions under which patterned states tend to be swept away by traveling waves even when the interaction is of excitatory-inhibitory type. The results also have implications for the numerical analysis of spatially discretized reaction-diffusion equations, where it is important to know whether solutions to the discretized equations converge to solutions to the continuum equation as the mesh size tends to zero.

37L60Lattice dynamics (infinite-dimensional dissipative systems)
34K20Stability theory of functional-differential equations
34K30Functional-differential equations in abstract spaces
34K60Qualitative investigation and simulation of models
82B20Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs
82B26Phase transitions (general)
35K99Parabolic equations and systems
Full Text: DOI