  We prove local existence and uniqueness of solutions for the one-dimensional
nonlinear Schr\"odinger (NLS) equations $iu_t + u_{xx} \pm |u|^2 u = 0$ in
classes of smooth functions that admit an asymptotic expansion at infinity in
decreasing powers of $x$. We show that an asymptotic solution differs from a
genuine solution by a Schwartz class function which solves a generalized
version of the NLS equation. The latter equation is solved by discretization
methods. The proofs closely follow previous work done by the author and others
on the Korteweg-De Vries (KdV) equation and the modified KdV equations.
