Scattering for small energy solutions of NLS with periodic potential in 1D

Abstract

Given H ≡ − ∂ x 2 + V ( x ) with V : R → R a smooth periodic potential, for μ ∈ R \\ { 0 } and p ⩾ 7 , we prove scattering for small solutions to i ∂ t u + H u = μ | u | p − 1 u , ( t , x ) ∈ R × R , u ( 0 ) = u 0 ∈ H 1 ( R ) . To cite this article: S. Cuccagna, N. Visciglia, C. R. Acad. Sci. Paris, Ser. I 347 (2009).