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 ∈ C¹ ([0, 1]), f(0) = f(1) = 0 < f(s) ∀ s ∈ (0, 1). Asymptotic expansions 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. 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.
SIAM Journal on Mathematical Analysis, 38(1), 233-258