Cauchy problem and dependency analysis for logarithmic Schr\"odinger equation on waveguide manifold

TL;DR

This paper investigates the Cauchy problem and dependency properties of the logarithmic Schrödinger equation on waveguide manifolds, introducing new methods due to the invalidity of traditional scaling arguments.

Contribution

It develops a novel approach to analyze y-dependence and stability for the logarithmic Schrödinger equation on product spaces, overcoming limitations of previous scaling-based methods.

Findings

Established new techniques for y-dependence analysis

Proved dynamical stability of solutions

Extended understanding of the Cauchy problem on waveguides

Abstract

In this paper, we develop a novel idea to study $y$-dependence for the logarithmic Schr\"odinger equation on $\mathbb{R}^d \times \mathbb{T}^n$. Unlike \cite{STNT2014}(Analysis \& PDE, 2014) and \cite{HHYL2024}(SIAM J. Math. Anal., 2024), the heart of the matter is that the scaling argument is invalid. Moreover, we also consider the Cauchy problem, which transforms the variational analysis into dynamical stability results.