Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics

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 is an element of Z, 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 c greater than or equal to c(min). Also, we show that traveling waves are unique up to a translation if f'(0) > 0 > f(l) and g'(0) > 0. The tails of traveling waves are also investigated.