Convergence to normal forms of integrable PDEs

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 neighbourhood 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.

References in corpus (4)

• Birkhoff coordinates for the Toda Lattice in the limit of infinitely many particles with an application to FPU
• Global Birkhoff coordinates for the periodic Toda lattice
• Long time stability of small finite gap solutions of the cubic Nonlinear Schrödinger equation on ${\mathbb{T}}^2$
• Tame majorant analyticity for the Birkhoff map of the defocusing Nonlinear Schrödinger equation on the circle

Cited by in corpus (2)

• About linearization of infinite-dimensional Hamiltonian systems
• From Klein-Gordon-Wave to Schrödinger-Wave: a Normal Form Approach