Asymptotic stability of the sine-Gordon kink
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913575865417728 |
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| author | Chen, Gong Luhrmann, Jonas |
| author_facet | Chen, Gong Luhrmann, Jonas |
| contents | We establish the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. Our proof consists of a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects combined with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major challenge is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the moving kink. Our analysis crucially relies on two remarkable null structures in the quadratic nonlinearities of the evolution equation for the radiation term and of the modulation equations.
The entire framework of our proof, including the systematic development of the distorted Fourier theory, is not specific to the sine-Gordon model and extends to many other asymptotic stability problems for moving solitons in relativistic scalar field theories on the line. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_07004 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic stability of the sine-Gordon kink Chen, Gong Luhrmann, Jonas Analysis of PDEs Mathematical Physics We establish the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. Our proof consists of a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects combined with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major challenge is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the moving kink. Our analysis crucially relies on two remarkable null structures in the quadratic nonlinearities of the evolution equation for the radiation term and of the modulation equations. The entire framework of our proof, including the systematic development of the distorted Fourier theory, is not specific to the sine-Gordon model and extends to many other asymptotic stability problems for moving solitons in relativistic scalar field theories on the line. |
| title | Asymptotic stability of the sine-Gordon kink |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2411.07004 |