We prove sharp $$L^\infty $$ decay and modified scattering for a one-dimensional dispersion-managed cubic nonlinear Schrödinger equation with small initial data chosen from weighted Sobolev space. Specifically, we work an averaged version of the NLS in strong dispersion management regime. The proof adapts techniques Hayashi Naumkin (Am. J. Math. 120(2):369–389, 1998) Kato Pusateri (Differ Integ...