Critical scattering for the nonlinear Schr\"odinger equation on waveguide manifolds

TL;DR

This paper proves small data scattering for the nonlinear Schrödinger equation on waveguide manifolds in critical spaces, overcoming key analytical challenges with anisotropic Strichartz estimates and novel interpolation techniques.

Contribution

It introduces an anisotropic framework and new proof techniques to establish small data scattering for NLS on waveguides, extending prior results to higher dimensions and broader nonlinearities.

Findings

Established small data scattering in critical spaces for NLS on waveguides.

Developed anisotropic Strichartz estimates with nearly unlimited endpoints.

Provided a new fixed point proof approach for the main result.

Abstract

We study the small data scattering problem in critical spaces for the nonlinear Schr\"odinger equation (NLS) on waveguide manifolds. Our work is primarily inspired by the recent paper of Kwak and Kwon \cite{KwakKwon} that established the local well-posedness of the periodic NLS with possibly non-algebraic nonlinearity. While we adopt a framework similar to \cite{KwakKwon} for our problem, two main obstacles prevent its direct adaptation to the waveguide setting. First, the classical Strichartz estimates for NLS in critical product spaces, introduced by Hani and Pausader, possess limited endpoints and are thus inapplicable to high-dimensional waveguides. Second, the crucial fractional arguments used in \cite{KwakKwon} rely on a well-known fractional derivative formula due to Strichartz, which admits only a Hilbert space-valued extension and is therefore incompatible with our model setting. To overcome these difficulties, we develop an anisotropic generalization of the framework in \cite{KwakKwon} using the anisotropic Strichartz estimates established by Tzvetkov and Visciglia, which allow for nearly unlimited endpoints. We also resolve several new challenges arising from the vector-valued and anisotropic nature of the model by employing novel interpolation techniques within Besov spaces. As a further novelty, we provide a new proof of the main result based on classical fixed point arguments, differing from the approximation methods used in \cite{KwakKwon}. Consequently, we settle the small data scattering problem in critical spaces for the NLS with arbitrary mass-supercritical nonlinearity on waveguide manifolds.