Uniqueness of travelling waves for nonlocal monostable equations

We consider a nonlocal analogue of the Fisher-KPP equation u_t = J * u - u + f(n), x ? R, f(0) = f(1) = 0, f > 0 on (0, 1), and its discrete counterpart u?_n = (J * u)_n - u _n + f(u_n), n ? Z, and show that travelling wave solutions of these equations that are bounded between 0 and 1 are unique up to translation. Our proof requires finding exact a priori asymptotics of a travelling wave. This we accomplish with the help of Ikehara's Theorem (which is a Tauberian theorem for Laplace transforms).