Transverse dependence and asymptotic dimensional reduction of ground states for the logarithmic Schrödinger equation on a waveguide manifold

Abstract

In this paper, we develop a novel approach to studying the $y$-dependence of ground states for the logarithmic Schrödinger equation on ${\mathbb{R}}^d\times \mathbb{T}$. Unlike in \cite{STNT2014} (Analysis \& PDE, 2014) and \cite{HH1} (SIAM J. Math. Anal., 2024), the power-law scaling argument does not transfer directly to the logarithmic setting. Instead, amplitude scaling gives an exact identity showing that transverse dependence is independent of the prescribed mass. We also discuss uniqueness for the Cauchy problem and conditional orbital stability.

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