Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics

Uniqueness of travelling waves for nonlocal monostable equations

“…In this work we extend the method of Diekmann and Kaper to obtain a similar, complete result such as the one in [5]. Let us discuss the differences between the two approaches.…”

“…Similar methods were used to establish existence of travelling waves of some other nonlocal models with monostable dynamics; see, e.g., [1], [4], [6], [13], [11], [14], [5].…”

“…The authors proved that travelling waves with noncritical speeds (i.e., c > c 0 ) are unique in the class of monotone solutions. Recently, Xinfu Chen and J.-S. Guo [5] obtained a complete uniqueness result for travelling waves for a generalized discrete version of the nonlocal monostable equation. To be more precise, they show that all such solutions, including nonmonotone waves and the one with the critical speed c 0 , that are bounded between 0 and 1 are unique up to translation.…”

“…The "strategy" of the two methods is similar: first one establishes an a priori asymptotic behaviour of a solution at 0, then two possible solutions are appropriately compared and shown to be the same. In [5], the authors first show that a solution u satisfies u < 0 and lim x→∞ u (x) u(x) = −α(c) (see below for the definition of α(c)). Then, assuming u 1 and u 2 are two solutions, they study…”

“…Both parts of our method are shorter than the corresponding ones in [5]. The main and crucial difference is that to obtain the exact asymptotics we construct a Laplace transform representation of a solution and then use the powerful Tauberian Ikehara's Theorem.…”