Modified scattering for nonlinear Schr\"odinger equations with long-range potentials

TL;DR

This paper establishes the existence of modified scattering states for nonlinear Schrödinger equations with long-range potentials and nonlinearities, including Coulomb potentials, by constructing unique global solutions with prescribed asymptotic profiles.

Contribution

It introduces a novel approach combining linear and nonlinear modifiers to prove modified scattering for NLS with both long-range nonlinearities and potentials, a first in the field.

Findings

Existence of modified wave operators for long-range potentials and nonlinearities.

Construction of asymptotic profiles using combined linear and nonlinear modifiers.

Extension of results to Coulomb potentials in 2D and 3D.

Abstract

We study the final state problem for the nonlinear Schr\"{o}dinger equation with a critical long-range nonlinearity and a long-range linear potential. Given a prescribed asymptotic profile which is different from the free evolution, we construct a unique global solution scattering to the profile. In particular, the existence of the modified wave operators is obtained for sufficiently localized small scattering data. The class of potential includes a repulsive long-range potential with a short-range perturbation, especially the positive Coulomb potential in two and three space dimensions. The asymptotic profile is constructed by combining Yafaev's type linear modifier [38] associated with the long-range part of the potential and the nonlinear modifier introduced by Ozawa [29]. Finally, we also show that one can replace Yafaev's type modifier by Dollard's type modifier under a slightly stronger decay assumption on the long-range potential. This is the first positive result on the modified scattering for the nonlinear Schr\"{o}dinger equation in the case when both of the nonlinear term and the linear potential are of long-range type.