Uniqueness and Asymptotics of Traveling Waves of Monostable Dynamics on Lattices

Established here is the uniquenes of solutions for the traveling wave problem cU � (x )= U (x+1)+U (x−1)−2U (x)+f (U (x)), x ∈ R, under the monostable nonlinearity: f ∈ C 1 ((0, 1)) ,f (0) = f (1) =0 <f (s) ∀ s ∈ (0, 1). Asymptotic expansions for U (x )a sx →± ∞, 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. 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.

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