# On the Korteweg-de Vries equation: convergent Birkhoff normal form

Bättig, D; Kappeler, T; Mityagin, B (1996). On the Korteweg-de Vries equation: convergent Birkhoff normal form. Journal of Functional Analysis, 140(2):335-358.

## Abstract

The Korteweg-de Vries (KdV) equation ∂tV(x,t)+∂x³V(x,t)−3∂xV(x,t)2=0 (x ∈ S1, t ∈ R) is a completely integrable Hamiltonian system of infinite dimension with phase space the Sobolev space HN(S1;R) (N≥1), Hamiltonian H(q):=∫S1(½(∂xq(x))2 + q(x)3)dx, and Poisson structure ∂/∂x. The function q≡0 is an elliptic fixed point. We prove that for any N≥1, the KdV equation (and thus the entire KdV hierarchy) admits globally defined real-analytic action-angle variables. As a consequence it follows that in a neighborhood of q≡0 in H1(S1;R), the KdV Hamiltonian H (and similarly any Hamiltonian in the KdV hierarchy) admits a convergent Birkhoff normal form; to the best of our knowledge this is the first such example in infinite dimension. Moreover, using the constructed action-angle variables, we analyze the regularity properties of the Hamiltonian vector field of the KdV equation.

The Korteweg-de Vries (KdV) equation ∂tV(x,t)+∂x³V(x,t)−3∂xV(x,t)2=0 (x ∈ S1, t ∈ R) is a completely integrable Hamiltonian system of infinite dimension with phase space the Sobolev space HN(S1;R) (N≥1), Hamiltonian H(q):=∫S1(½(∂xq(x))2 + q(x)3)dx, and Poisson structure ∂/∂x. The function q≡0 is an elliptic fixed point. We prove that for any N≥1, the KdV equation (and thus the entire KdV hierarchy) admits globally defined real-analytic action-angle variables. As a consequence it follows that in a neighborhood of q≡0 in H1(S1;R), the KdV Hamiltonian H (and similarly any Hamiltonian in the KdV hierarchy) admits a convergent Birkhoff normal form; to the best of our knowledge this is the first such example in infinite dimension. Moreover, using the constructed action-angle variables, we analyze the regularity properties of the Hamiltonian vector field of the KdV equation.

## Citations

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## Additional indexing

Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 1996 29 Nov 2010 16:28 05 Apr 2016 13:27 Elsevier 0022-1236 10.1006/jfan.1996.0111 http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0868.35099

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