Dependence of the nonlinear Schr{\"o}dinger flow upon the nonlinearity

TL;DR

This paper studies how solutions to the defocusing nonlinear Schrödinger equation depend on the nonlinearity power, focusing on global behavior, limits, long-range effects, and scattering operator continuity.

Contribution

It provides a detailed analysis of the solution dependence on nonlinearity power, including limits, long-range effects, and scattering operator continuity in the energy-subcritical case.

Findings

Analyzes the limit as the power tends to one and its connection to the logarithmic Schrödinger equation.

Describes the behavior when long-range effects are present.

Establishes the continuity of the scattering operator in the short-range case.

Abstract

We consider the defocusing nonlinear Schr{\"o}dinger equation in the energy-subcritical case, and investigate the dependence of the solution upon the power of the nonlinearity. Special attention is paid to the global in time description. The main three aspects addressed, in the decreasing order of difficulty, are the limit when the total power tends to one, along with the connection with the logarithmic Schr{\"o}dinger equation, the description when long range effects may be present, and the continuity of the scattering operator in the short range case. This text resumes the presentation given by the first author at {\'E}cole polytechnique for the Laurent Schwartz seminar, in May 2026.