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Zbl 1152.37033
Chen, Xinfu; Guo, Jong-Shenq; Wu, Chin-Chin
Traveling waves in discrete periodic media for bistable dynamics.
(English)
[J] Arch. Ration. Mech. Anal. 189, No. 2, 189-236 (2008). ISSN 0003-9527; ISSN 1432-0673/e

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 $$\dot u_i=\sum_ka_{i,j}u_{i+k}+f_i(u_i),\qquad t>0,\,i\in{\mathbb Z}$$ 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.
[Christian Pötzsche (München)]
MSC 2000:
*37L60 Lattice dynamics
35K55 Nonlinear parabolic equations
37L15 Stability problems of infinite-dimensional Hamiltonian systems
35K15 Second order parabolic equations, initial value problems
35B10 Periodic solutions of PDE

Keywords: lattice dynamics; traveling waves; bistable dynamics; discrete periodic media; global stability

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