Modified wave operators for the defocusing cubic nonlinear Schr\"odinger equation in one space dimension with large scattering data

TL;DR

This paper constructs modified wave operators for the defocusing cubic NLS in one dimension with large scattering data, using a new linearization approach and energy estimates without relying on integrability.

Contribution

Introduces a novel formulation based on linearization around an asymptotic profile, enabling construction of wave operators without size restrictions or integrability assumptions.

Findings

Successfully constructs modified wave operators for large scattering data.

Provides an improved explicit upper bound for scattering data in the focusing case.

Develops a new energy method applicable to long-range potentials and short-range perturbations.

Abstract

In the present paper, we construct modified wave operators for the defocusing cubic nonlinear Schr\"odinger equation (NLS) in one space dimension without size restriction on scattering data. In the proof, we introduce a new formulation of the problem based on the linearization of the NLS around a prescribed asymptotic profile. For the linearized equation which is a system of Schr\"odinger equations with non-symmetric, time-dependent long-range potentials, we show a modified energy identity, as well as an associated energy estimate, which allow us to apply a simple energy method to construct the modified wave operators. As a byproduct, we also obtain in the focusing case an improved explicit upper bound for the size of scattering data to ensure the existence of modified wave operators. Our argument relies neither on the complete integrability nor on the framework of analytic function spaces, and also works for short-range perturbations of the cubic nonlinearity.