Second Order Spectral Estimates and Symmetry Breaking for Rotating Wave Solutions

TL;DR

This paper analyzes the spectral properties of a differential operator arising from rotating wave solutions of a nonlinear wave equation on a disk, revealing how spectrum structure and symmetry breaking depend on arithmetic properties of a key parameter.

Contribution

It introduces a detailed spectral analysis of the elliptic-hyperbolic operator $L_eta$, classifies its spectrum based on rationality of a parameter, and extends symmetry breaking results for ground state solutions.

Findings

Spectrum structure depends on the rationality of $\sigma$.

Explicit estimates for Bessel function zeros are provided.

Existence and symmetry breaking of solutions are characterized.

Abstract

We consider rotating wave solutions of the nonlinear wave equation \[ \left\{ \begin{aligned} \partial_{t}^2 v - \Delta v + m v & = |v|^{p-2} v \quad && \text{in $\mathbb{R} \times \textbf{B}$} \\ v & = 0 && \text{on $\mathbb{R} \times \partial \textbf{B}$} \end{aligned} \right. \] for $2<p<\infty$, $m \in \mathbb{R}$ on the unit disk $\textbf{B} \subset \mathbb{R}^2$. This leads to the study of a reduced equation involving the elliptic-hyperbolic operator $L_\alpha = -\Delta + \alpha^2 \partial_{\theta}^2$ with $\alpha>1$. We find that the structure of the spectrum of $L_\alpha$ strongly depends on the quantity \[ \sigma = \frac{\pi}{\sqrt{\alpha^2- 1} - \arccos \frac{1}{\alpha}} > 0 . \] By giving precise estimates for certain sequences of Bessel function zeros, we can classify the spectrum for all $\alpha>1$ such that $\sigma$ is rational and further find that the existence of accumulation points explicitly depends on arithmetic properties of $\sigma$. Using these characterizations, we deduce existence and symmetry breaking results for ground state solutions of the reduced equation, extending known results.