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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2501.00109 |
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| _version_ | 1866915086737604608 |
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| author | Kübler, Joel |
| author_facet | Kübler, Joel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_00109 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Second Order Spectral Estimates and Symmetry Breaking for Rotating Wave Solutions Kübler, Joel Analysis of PDEs 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. |
| title | Second Order Spectral Estimates and Symmetry Breaking for Rotating Wave Solutions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.00109 |