Finite energy well-posedness for nonlinear Schr\"odinger equations with non-vanishing conditions at infinity

TL;DR

This paper establishes local and global well-posedness results for nonlinear Schrödinger equations with non-vanishing conditions at infinity in 2D and 3D, extending the energy space theory beyond vanishing boundary conditions.

Contribution

It proves local well-posedness in the energy space for non-vanishing conditions and demonstrates global well-posedness under specific physical and mathematical assumptions, including stability conditions.

Findings

Local well-posedness in energy space for 2D and 3D cases.

Global well-posedness under stability and small initial data assumptions.

Extension of energy space theory to non-vanishing boundary conditions.

Abstract

Relevant physical phenomena are described by nonlinear Schr\"odinger equations with non-vanishing conditions at infinity. This paper investigates the respective 2D and 3D Cauchy problems. Local well-posedness in the 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