Existence of traveling waves for Lipschitz discrete dynamics. Monostable case as a limit of bistable cases

We study discrete monostable dynamics with general Lipschitz non-linearities. This includes also degenerate non-linearities. In the positive monostable case, we show the existence of a branch of traveling waves solutions for velocities c ≥ c+, with non existence of solutions for c < c+. We also give certain sufficient conditions to insure that c+ ≥ 0 and we give an example when c+ < 0. We as well prove a lower bound of c+, precisely we show that c+ ≥ c∗, where c∗ is associated to a linearized problem at infinity. On the other hand, under a KPP condition we show that c+ ≤ c∗. We also give an example where c+ > c∗. This model of discrete dynamics can be seen as a generalized Frenkel-Kontorova model for which we can also add a driving force parameter σ. We show that σ can vary in an interval [σ−, σ+]. For σ ∈ (σ−, σ+) this corresponds to a bistable case, while for σ = σ+ this is a positive monostable case, and for σ = σ− this is a negative monostable case. We study the velocity function c = c(σ) as σ varies in [σ−, σ+]. In particular for σ = σ+ (resp. σ = σ−), we find vertical branches of traveling waves solutions with c ≥ c+ (resp. c ≤ c−). Our method of proof is new and relies on viscosity solutions. Moreover, the monostable case with c = c+ is seen advantageously as a limit situation of the bistable case. For c >> 1, the traveling waves are constructed as perturbations of solutions of an associated ODE. Finally to fill the gap between c = c+ and large c, we use certain hull functions that are associated to correctors of a homogenization problem.