We consider the Cauchy problem for the nonlinear Schrödinger equation on ℝ², , λ∈ℝ, σ>0. We introduce new functional spaces over which the initial value problem is well-posed. Their construction is based on spatial plane waves. These spaces contain and do not lie within . We prove several global well-posedness and stability results over these new spaces, including a new global well-posedness result of H¹ solutions with indefinitely large H¹ and L² norms. Some of these results are proved using a new functional transform, the 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.