Dynamics of the sine-Gordon equation on tadpole graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911442726289408 |
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| author | Pava, Jaime Angulo Plaza, Ramón G. |
| author_facet | Pava, Jaime Angulo Plaza, Ramón G. |
| contents | This work studies the dynamics of solutions to the sine-Gordon equation posed on a tadpole graph $G$ and endowed with boundary conditions at the vertex of $δ$-type. The latter generalize conditions of Neumann-Kirchhoff type. The purpose of this analysis is to establish an instability result for a certain family of stationary solutions known as \emph{single-lobe kink state profiles}, which consist of a periodic, symmetric, concave stationary solution in the finite (periodic) lasso of the tadpole, coupled with a decaying kink at the infinite edge of the graph. It is proved that such stationary profile solutions are linearly (and nonlinearly) unstable under the flow of the sine-Gordon model on the graph. The extension theory of symmetric operators, Sturm-Liouville oscillation results and analytic perturbation theory of operators are fundamental ingredients in the stability analysis. The local well-posedness of the sine-Gordon model in an appropriate energy space is also established. The theory developed in this investigation constitutes the first stability result of stationary solutions to the sine-Gordon equation on a tadpole graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16726 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dynamics of the sine-Gordon equation on tadpole graphs Pava, Jaime Angulo Plaza, Ramón G. Analysis of PDEs 35Q51, 35J61, 35B35, 47E05 This work studies the dynamics of solutions to the sine-Gordon equation posed on a tadpole graph $G$ and endowed with boundary conditions at the vertex of $δ$-type. The latter generalize conditions of Neumann-Kirchhoff type. The purpose of this analysis is to establish an instability result for a certain family of stationary solutions known as \emph{single-lobe kink state profiles}, which consist of a periodic, symmetric, concave stationary solution in the finite (periodic) lasso of the tadpole, coupled with a decaying kink at the infinite edge of the graph. It is proved that such stationary profile solutions are linearly (and nonlinearly) unstable under the flow of the sine-Gordon model on the graph. The extension theory of symmetric operators, Sturm-Liouville oscillation results and analytic perturbation theory of operators are fundamental ingredients in the stability analysis. The local well-posedness of the sine-Gordon model in an appropriate energy space is also established. The theory developed in this investigation constitutes the first stability result of stationary solutions to the sine-Gordon equation on a tadpole graph. |
| title | Dynamics of the sine-Gordon equation on tadpole graphs |
| topic | Analysis of PDEs 35Q51, 35J61, 35B35, 47E05 |
| url | https://arxiv.org/abs/2502.16726 |