  We prove sharp $L^\infty$ decay and modified scattering for a one-dimensional
dispersion-managed cubic nonlinear Schr\"odinger equation with small initial
data chosen from a weighted Sobolev space. Specifically, we work with an
averaged version of the dispersion-managed NLS in the strong dispersion
management regime. The proof adapts techniques from Hayashi-Naumkin and
Kato-Pusateri, which established small-data modified scattering for the
standard $1d$ cubic NLS.
