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.

About linearization of infinite-dimensional Hamiltonian systems

“…Furthermore, in general, the notion of formal normal form and formal change of variables should be clarified (for instance if one defines formal polynomials and formal power series it is not in general true that this space has a Poisson algebra structure). Neverteless, in some very peculiar situation, this problem can be handled [BS20]. † † Research of L. Stolovitch was supported by the French government, through the UCAJEDI Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-01.…”

Complete Integrability of Diffeomorphisms and Their Local Normal Forms

“…Later, Eliasson, Ito, Stolovitch, Zung to name a few, generalized or improved Vey's theorem in different aspects including non-Hamiltonian setting [8,9,12,13,19,20,29]. Such a concept has also been recently developed in the context of PDE's as infinite-dimensional dynamical systems [4,15,16]. In a different context of global dynamics, a notion of "integrable maps" has been devised relative to long-time behavior of their orbits and their complexity [22,23].…”