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

Mohammad Al Haj (1) , Régis Monneau (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

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− ).

Mots clés: Perron's method; Traveling waves; degenerate monostable non-linearity; KPP non-linearity; bistable non-linearity; Frenkel-Kontorova model; viscosity solutions; Perron's method.

Domaines: Equations aux dérivées partielles [math.AP]

Mohammad Al Haj, Régis Monneau. Existence of traveling waves for Lipschitz discrete dynamics. Monostable case as a limit of bistable cases. 2014. ⟨hal-00921710v3⟩