  We consider the Cauchy problem for the nonlinear Schr\"odinger equation on
$\mathbb{R}^2$, $iu_t + u_{xx} + u_{yy} + \lambda|u|^\sigma u =0$, $\lambda\in
\mathbb{R}$, $\sigma>0$. We introduce new functional spaces over which the
initial value problem is well-posed. Their construction is based on
\textit{spatial plane waves} (cf. arXiv:1510.08745). These spaces contain
$H^1(\mathbb{R}^2)$ and do not lie within $L^2(\mathbb{R}^2)$. We prove several
global well-posedness and stability results over these new spaces, including a
new global well-posedness result of $H^1$ solutions with indefinitely large
$H^1$ and $L^2$ norms. Some of these results are proved using a new functional
transform, the \textit{plane wave transform}. We develop a suitable theory for
this transform, prove several properties and solve classical linear PDE's with
it, highlighting its wide range of application.
