In an infinite dimensional Hilbert space we consider a family of commuting analytic vector fields vanishing at the origin and which are nonlinear perturbations of some fundamental linear vector fields. We prove that one can construct by the method of Poincaré normal form a local analytic coordinate transformation near the origin transforming the family into a normal form. The result applies to the KdV and NLS equations and to the Toda lattice with periodic boundary conditions. One gets existence of Birkhoff coordinates in a neighborhood of the origin. The proof is obtained by directly estimating, in an iterative way, the terms of the Poincaré normal form and of the transformation to it, through a rapid convergence algorithm.
Convergence to Normal Forms of Integrable PDEs / D. Bambusi, L. Stolovitch. - In: COMMUNICATIONS IN MATHEMATICAL PHYSICS. - ISSN 0010-3616. - 376:2(2020), pp. 1441-1470. [10.1007/s00220-019-03661-8]
Convergence to Normal Forms of Integrable PDEs
D. Bambusi
Primo
;
2020
Abstract
In an infinite dimensional Hilbert space we consider a family of commuting analytic vector fields vanishing at the origin and which are nonlinear perturbations of some fundamental linear vector fields. We prove that one can construct by the method of Poincaré normal form a local analytic coordinate transformation near the origin transforming the family into a normal form. The result applies to the KdV and NLS equations and to the Toda lattice with periodic boundary conditions. One gets existence of Birkhoff coordinates in a neighborhood of the origin. The proof is obtained by directly estimating, in an iterative way, the terms of the Poincaré normal form and of the transformation to it, through a rapid convergence algorithm.File | Dimensione | Formato | |
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Bambusi-Stolovitch2020_Article_ConvergenceToNormalFormsOfInte.pdf
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