Uniqueness and asymptotics of traveling waves of monostable dynamics on lattices

Abstract. Established here is the uniquenes of solutions for the traveling wave prob-lem cU ′(x) = U(x + 1) + U(x − 1) − 2U(x) + f(U(x)), x ∈ R, under the monostable non-linearity: f ∈ C1([0, 1]), f(0) = f(1) = 0 < f(s) ∀ s ∈ (0, 1). Asymptotic expan-sions for U(x) as x → ±∞, accurate enough to capture the translation differences, are also derived and rigorously verified. These results complement earlier existence and partial uniqueness/stability results in the literature (e.g. [3, 4, 18]). New tools are also developed to deal with the degenerate case f ′(0)f ′(1) = 0, about which is the main concern of this article