Regularity of nonvanishing – at infinity or at the boundary – solutions of the defocusing nonlinear Schrödinger equation

Considering the defocusing nonlinear Schrödinger equation (NLSE) in generic (bounded or unbounded) open sets U⊆Rn for n = 1, 2, and 3, we prove the regularity of weak, non-vanishing solutions at infinity or at the boundary of U. Our approach is based on suitably defined extension operators, along with a priori estimates for regular functions, under certain assumptions on the smoothness of the boundary. The results cover physically significant classes of solutions, as dark-solitons and compacton waveforms, when the notion of such solutions is extended in higher-dimensional set-ups.