Existence of traveling waves for Lipschitz discrete dynamics. Monostable case as a limit of bistable cases
Mohammad Al Haj, Régis Monneau · 2014
Abstract: 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∗. 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.