Uniqueness of travelling waves for nonlocal monostable equations

We consider a nonlocal analogue of the Fisher-KPP equation \[ u t = J ∗ u − u + f ( u ) , x ∈ R , f ( 0 ) = f ( 1 ) = 0 , f > 0 on ( 0 , 1 ) , u_t =J*u-u+f(u),~x\in R,~f(0)=f(1)=0,~f>0 ~\textrm {on}~(0,1), \] and its discrete counterpart u ˙ n = ( J ∗ u ) n − u n + f ( u n ) {\dot u}_n =(J*u)_n -u_n +f(u_n ) , n ∈ Z n\in Z , and show that travelling wave solutions of these equations that are bounded between 0 0 and 1 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).