Second Order Spectral Estimates and Symmetry Breaking for Rotating Wave Solutions

TL;DR

The paper analyzes rotating wave solutions of the nonlinear Klein–Gordon equation on the unit disk, reducing the problem to a mixed elliptic-hyperbolic operator $L_\alpha = -\Delta + \alpha^2 \partial_{\theta}^2$ with $\alpha>1$ and spectral quantity $\sigma = \pi/(\sqrt{\alpha^2-1} - \arccos(1/\alpha))$. It develops refined asymptotics for the zeros of Bessel functions, establishing a precise second-order expansion for $j_{\ell,k}$ via $\zeta_x$, and shows how arithmetic properties of rational $\sigma$ govern spectrum structure, including absence of accumulation points under certain divisibility conditions. These spectral insights enable existence and symmetry-breaking results for ground states of the reduced equation, extending prior work to the elliptic-hyperbolic regime. The combined analytic and variational approach highlights a deep connection between spectral arithmetic and nonlinear wave symmetry on bounded domains, with implications for nonradial ground states and rotating-wave stability.

Abstract

We consider rotating wave solutions of the nonlinear wave equation \[ \left\{ \begin{aligned} \partial_{t}^2 v - Δv + m v & = |v|^{p-2} v \quad && \text{in $\mathbb{R} \times \textbf{B}$} \\ v & = 0 && \text{on $\mathbb{R} \times \partial \textbf{B}$} \end{aligned} \right. \] 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 $L_α= -Δ+ α^2 \partial_θ^2$ with $α>1$. We find that the structure of the spectrum of $L_α$ strongly depends on the quantity \[ σ= \fracπ{\sqrt{α^2- 1} - \arccos \frac{1}α} > 0 . \] By giving precise estimates for certain sequences of Bessel function zeros, we can classify the spectrum for all $α>1$ such that $σ$ is rational and further find that the existence of accumulation points explicitly depends on arithmetic properties of $σ$. Using these characterizations, we deduce existence and symmetry breaking results for ground state solutions of the reduced equation, extending known results.

Theorems & Definitions (45)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Theorem 2.2
• Lemma 2.3
• Remark 2.4
• Lemma 2.5
• Lemma 2.6