Abstract
Existence of traveling wave solutions for some lattice differential equations is investigated. We prove that there exists c*>0 such that for each c≥c*, the systems under consideration admit monotonic nondecreasing traveling waves.
Highlights
Consider the following lattice differential equation ( ) = u i (v ui+1 − 2ui ) + ui−1 − f ui, ( Bu )i + αvi, i ∈ Ζ,v i = −σ vi + βui, i ∈ Ζ, (1.1)where v, σ are positive constants, αβ > 0, f is aC2 -function, and ( Bu=) i ui+1 − ui .Lattice dynamical systems occur in a wide variety of applications, and a lot of studies have been done, e.g., see [1]-[4].A pair of solutions { }ui ∞, i=−∞ { }v ∞ i i=−∞ of (1.1)
Lattice dynamical systems occur in a wide variety of applications, and a lot of studies have been done, e.g., see pair of solutions
The main result of this paper is Theorem 1.1
Summary
Is called a traveling wave solution with wave speed c > 0 if there exist functions U ,V : R → R such that = ui U (i + ct ) , =vi V (i + ct ) with (U (−∞) ,V (−∞)) = (U− ,V− ) and (U (+∞) ,V (+∞)) = (U+ ,V+ ) . Let ξ = i + ct , note that (1.1) has a pair of traveling wave solutions if and only if U , V satisfy the functional differential equation cU = (ξ ) v (U (ξ +1) − 2U (ξ ) + U (ξ −1)) − f (U (ξ ), BU (ξ )) + αV (ξ ), (2016) Existence of Traveling Waves in Lattice Dynamical Systems.
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