Reducible KAM Tori for the Degasperis-Procesi equation
We develop KAM theory close to an elliptic fixed point for quasi-linear Hamiltonian perturbations of the dispersive Degasperis–Procesi equation on the circle. The overall strategy in KAM theory for quasi-linear PDEs is based on Nash–Moser nonlinear iteration, pseudo differential calculus and normal...
| Autores: | , , |
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| Tipo de documento: | artigo |
| Data de publicação: | 2020 |
| País: | España |
| Recursos: | Universitat Politècnica de Catalunya (UPC) |
| Repositório: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglês |
| OAI Identifier: | oai:upcommons.upc.edu:2117/332387 |
| Acesso em linha: | https://hdl.handle.net/2117/332387 https://dx.doi.org/10.1007/s00220-020-03788-z |
| Access Level: | Acceso aberto |
| Palavra-chave: | Hamiltonian systems Sistemes hamiltonians Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Resumo: | We develop KAM theory close to an elliptic fixed point for quasi-linear Hamiltonian perturbations of the dispersive Degasperis–Procesi equation on the circle. The overall strategy in KAM theory for quasi-linear PDEs is based on Nash–Moser nonlinear iteration, pseudo differential calculus and normal form techniques. In the present case the complicated symplectic structure, the weak dispersive effects of the linear flow and the presence of strong resonant interactions require a novel set of ideas. The main points are to exploit the integrability of the unperturbed equation, to look for special wave packet solutions and to perform a very careful algebraic analysis of the resonances. Our approach is quite general and can be applied also to other 1d integrable PDEs. We are confident for instance that the same strategy should work for the Camassa–Holm equation. |
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