Abstract
This paper is concerned with surface waves, solutions of hyperbolic nonlinear boundary value problems. We construct BKW solutions in the weakly nonlinear regime with infinite expansion in powers of ε. We rigorously justify this expansion, constructing exact solutions, which admit the asymptotic expansions. We also show that the solution is not necessarily localized at the order 𝒪(ε∞) in the interior, even if the data are.