Finite energy well-posedness for nonlinear Schrödinger equations with non-vanishing conditions at infinity

Abstract

Relevant physical phenomena are described by nonlinear Schrödinger equations with non-vanishing conditions at infinity. This paper investigates the respective 2D and 3D Cauchy problems. Local well-posedness in the (curved) energy space, for energy-subcritical nonlinearities merely satisfying Kato-type assumptions, is proven, providing the analogue of the well-established local $H^1$-theory for solutions vanishing at infinity. The critical nonlinearity will be simply a byproduct of our analysis and the existing literature. Under an assumption that prevents the onset of a Benjamin–Feir type instability, global well-posedness in the energy space is proven for: (a) non-negative Hamiltonians, (b) sign-indefinite Hamiltonians under additional assumptions on the zeros of the nonlinearity, (c) generic nonlinearities and small initial data. The cases (b) and (c) only concern the 3D case.