Breather solutions for semilinear wave equations
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915292776497152 |
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| author | Henninger, Julia Ohrem, Sebastian Reichel, Wolfgang |
| author_facet | Henninger, Julia Ohrem, Sebastian Reichel, Wolfgang |
| contents | We prove existence of real-valued, time-periodic and spatially localized solutions (breathers) of semilinear wave equations $V(x)u_{tt} - u_{xx} = Γ(x) |u|^{p-1} u$ on $\mathbb{R}^2$ for all values of $p\in (1,\infty)$. Using tools from the calculus of variations our main result provides breathers as ground states of an indefinite functional under suitable conditions on $V, Γ$ beyond the limitations of pure $x$-periodicity. Such an approach requires a detailed analysis of the wave operator acting on time-periodic functions. Hence a generalization of the Floquet-Bloch theory for periodic Sturm-Liouville operators is needed which applies to perturbed periodic operators. For this purpose we develop a suitable functional calculus for the weighted operator $-\frac{1}{V(x)}\frac{\mathrm{d}^2}{\mathrm{d}x^2}$ with an explicit control of its spectral measure. Based on this we prove embedding theorems from the form domain of the wave operator into $L^q$-spaces, which is key to controlling nonlinearities. We complement our existence theory with explicit examples of coefficient functions $V$ and temporal periods $T$ which support breathers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_13336 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Breather solutions for semilinear wave equations Henninger, Julia Ohrem, Sebastian Reichel, Wolfgang Analysis of PDEs Primary: 35L71, 49J35, Secondary: 35B10, 34L05 We prove existence of real-valued, time-periodic and spatially localized solutions (breathers) of semilinear wave equations $V(x)u_{tt} - u_{xx} = Γ(x) |u|^{p-1} u$ on $\mathbb{R}^2$ for all values of $p\in (1,\infty)$. Using tools from the calculus of variations our main result provides breathers as ground states of an indefinite functional under suitable conditions on $V, Γ$ beyond the limitations of pure $x$-periodicity. Such an approach requires a detailed analysis of the wave operator acting on time-periodic functions. Hence a generalization of the Floquet-Bloch theory for periodic Sturm-Liouville operators is needed which applies to perturbed periodic operators. For this purpose we develop a suitable functional calculus for the weighted operator $-\frac{1}{V(x)}\frac{\mathrm{d}^2}{\mathrm{d}x^2}$ with an explicit control of its spectral measure. Based on this we prove embedding theorems from the form domain of the wave operator into $L^q$-spaces, which is key to controlling nonlinearities. We complement our existence theory with explicit examples of coefficient functions $V$ and temporal periods $T$ which support breathers. |
| title | Breather solutions for semilinear wave equations |
| topic | Analysis of PDEs Primary: 35L71, 49J35, Secondary: 35B10, 34L05 |
| url | https://arxiv.org/abs/2505.13336 |