# 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)
Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice. (English) Zbl 0917.34052
Summary: The authors consider infinite systems of ODEs on the two-dimensional integer lattice, given by a bistable scalar ODE at each point, with a nearest neighbor coupling between lattice points. For a class of ideal nonlinearities, they obtain traveling wave solutions in each direction ${e}^{i\theta }$, and explore the relation between the wave speed $c$, the angle $\theta$, and the detuning parameter $a$ of the nonlinearity. Of particular interest is the phenomenon of “propagation failure”, and the authors study how the critical value $a={a}^{*}\left(\theta \right)$ depends on $\theta$, where ${a}^{*}\left(\theta \right)$ is defined as the value of the parameter $a$ at which propagation failure (that is, wave speed c=0) occurs. The authors show that ${a}^{*}:ℝ\to ℝ$ is continuous at each point $\theta$ where $tan\theta$ is irrational, and is discontinuous where $tan\theta$ is rational or infinite.
##### MSC:
 34G20 Nonlinear ODE in abstract spaces 34A35 ODE of infinite order 35K57 Reaction-diffusion equations 74J99 Waves (solid mechanics)