Analysis of the sine-Gordon equation with a nonlinear $δ$-potential
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908988059156480 |
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| author | Moroni, Sergio Plaza, Ramón G. |
| author_facet | Moroni, Sergio Plaza, Ramón G. |
| contents | This paper is devoted to the analysis of the following nonlinear wave equation \[ u_{tt} - u_{xx} + (1 + qδ_0(x)) \sin u = 0, \] where $δ_0 = δ_0(x)$ is the Dirac delta function centered at the origin and $q \in \mathbb{R}$ is a constant. Equations of this form arise in the study of propagating solitons in the presence of a localized inhomogeneity. It is proved that the Cauchy problem for this equation is globally well-posed in the energy space $H^1_{\sin} \times L^2$. A complete characterization of stationary waves in the energy space, based on the parameter $q$, is also provided. Finally, a criterion to determine the stability or instability of the stationary waves, which depends upon the sign of the parameter $q$, is established. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21185 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Analysis of the sine-Gordon equation with a nonlinear $δ$-potential Moroni, Sergio Plaza, Ramón G. Analysis of PDEs 35Q51, 35J61, 47E05 This paper is devoted to the analysis of the following nonlinear wave equation \[ u_{tt} - u_{xx} + (1 + qδ_0(x)) \sin u = 0, \] where $δ_0 = δ_0(x)$ is the Dirac delta function centered at the origin and $q \in \mathbb{R}$ is a constant. Equations of this form arise in the study of propagating solitons in the presence of a localized inhomogeneity. It is proved that the Cauchy problem for this equation is globally well-posed in the energy space $H^1_{\sin} \times L^2$. A complete characterization of stationary waves in the energy space, based on the parameter $q$, is also provided. Finally, a criterion to determine the stability or instability of the stationary waves, which depends upon the sign of the parameter $q$, is established. |
| title | Analysis of the sine-Gordon equation with a nonlinear $δ$-potential |
| topic | Analysis of PDEs 35Q51, 35J61, 47E05 |
| url | https://arxiv.org/abs/2604.21185 |