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Traveling waves in discrete periodic media for bistable dynamics. (English) Zbl 1152.37033

The main purpose of this paper is to introduce a general framework for the study of traveling waves in discrete periodic media. It is concerned with the existence, uniqueness, and global stability of traveling waves in discrete periodic media for an infinite system of ordinary differential equations

${\stackrel{˙}{u}}_{i}=\sum _{k}{a}_{i,j}{u}_{i+k}+{f}_{i}\left({u}_{i}\right),\phantom{\rule{2.em}{0ex}}t>0,\phantom{\rule{0.166667em}{0ex}}i\in ℤ$

exhibiting bistable dynamics. It is assumed that ${a}_{i,k},{f}_{i}$ are periodic in $i$ and that ordered steady states exist. Moreover, the coefficients ${a}_{i,k}$ are supposed to fulfill a (discrete) ellipticity condition, a non-decoupledness condition and that one has a finite range interaction in the sense that ${a}_{i,k}$ vanishes for large absolute values of $k$.

The main tools used to prove the uniqueness and asymptotic stability of traveling waves are the comparison principle, spectrum analysis based on the Krein-Rutman theorem, and constructions of super/subsolutions. To prove the existence of traveling waves, the system is converted to an integral equation which is common in the study of monostable dynamics but quite rare in the study of bistable dynamics.

##### MSC:
 37L60 Lattice dynamics (infinite-dimensional dissipative systems) 35K55 Nonlinear parabolic equations 37L15 Stability problems of infinite-dimensional dissipative systems 35K15 Second order parabolic equations, initial value problems 35B10 Periodic solutions of PDE