Global well-posedness and scattering in weighted space for nonlinear Schr\"{o}dinger equations below the Strauss exponent without gauge-invariance

TL;DR

This paper proves global well-posedness and scattering for small solutions of nonlinear Schrödinger equations with general homogeneous nonlinearities below the Strauss exponent, including non-gauge-invariant cases, in weighted spaces.

Contribution

It introduces a new formulation to handle non-gauge-invariant nonlinearities where standard methods fail, extending GWP and scattering results to broader classes of NLS.

Findings

Established GWP and scattering for small data in weighted spaces

Included nonlinearities below the Strauss exponent

Handled non-gauge-invariant nonlinearities with a new formulation

Abstract

In this paper, we consider the nonlinear Schr\"{o}dinger equation (NLS) with a general homogeneous nonlinearity in dimensions up to three. We assume that the degree (i.e., power) of the nonlinearity is such that the equation is mass-subcritical and short-range. We establish global well-posedness (GWP) and scattering for small data in the standard weighted space for a class of homogeneous nonlinearities, including non-gauge-invariant ones. Additionally, we include the case where the degree is less than or equal to the Strauss exponent. When the nonlinearity is not gauge-invariant, the standard Duhamel formulation fails to work effectively in the weighted Sobolev space; for instance, the Duhamel term may not be well-defined as a Bochner integral. To address this issue, we introduce an alternative formulation that allows us to establish GWP and scattering, even in the presence of poor time continuity of the Duhamel term.