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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 theinitial 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 severalglobal well-posedness and stability results over these new spaces, including anew 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 functionaltransform, the \textit{plane wave transform}. We develop a suitable theory forthis transform, prove several properties and solve classical linear PDE's withit, highlighting its wide range of application.

Spatial plane waves for the nonlinear Schr\"odinger equation: local existence and stability results