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Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics. (English) Zbl 1086.34011

Summary: We study traveling waves of a discrete system

u ˙ j =g(u j+1 )+g(u j-1 )-2g(u j )+f(u j ),j,

where f and g are Lipschitz continuous with g increasing and f monostable, i.e., f(0)=f(1)=0 and f>0 on (0,1). We show that there is a positive c min such that a traveling wave of speed c exists if and only if cc min . Also, we show that traveling waves are unique up to a translation if f ' (0)>0>f ' (1) and g ' (0)>0. The tails of traveling waves are also investigated.

34A35ODE of infinite order
34C37Homoclinic and heteroclinic solutions of ODE
35K55Nonlinear parabolic equations
37L60Lattice dynamics (infinite-dimensional dissipative systems)