Almost sure well-posedness and orbital stability for Schr\"odinger equation with potential

TL;DR

This paper establishes almost sure well-posedness and the first orbital stability results for a nonlinear Schrödinger equation with potential in four dimensions, using advanced harmonic analysis techniques.

Contribution

It provides the first orbital stability result for the Schrödinger equation with potential, extending well-posedness theory to a probabilistic setting in four dimensions.

Findings

Proved almost sure well-posedness for the model.

Established orbital stability of solutions.

Developed new analytical techniques involving Strichartz and smoothing spaces.

Abstract

In this paper, we study the almost sure well-posedness theory and orbital stability for the nonlinear Schr\"odinger equation with potential \begin{equation*} \left\{\begin{array}{l} i \partial_t u+\Delta u-V(x)u+|u|^{2}u=0,\ (x, t) \in \mathbb{R}^4 \times \mathbb{R}, \\ \left.u\right|_{t=0}=f \in H ^s(\mathbb{R}^4), \end{array}\right. \end{equation*} where $\frac{1}{3}<s<1$ and $V(x):\mathbb{R}^4\rightarrow \mathbb{R}$ satisfies appropriate conditions. The main idea in the proofs is based on Strichartz spaces as well as variants of local smoothing, inhomogeneous local smoothing and maximal function spaces. To our best knowledge, this is the first orbital stability result for this model.