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 is an element of R, under the monostable nonlinearity: f is an element of C-1([0, 1]), f(0) = f( 1) = 0 < f(s) for all s is an element of (0, 1). Asymptotic expansions for U(x) as x --> +/-infinity, 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.