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Title of article :
On the Korteweg–de Vries Equation: Convergent Birkhoff Normal Form
Author/Authors :
Bنttig، نويسنده , , D and Kappeler، نويسنده , , T and Mityagin، نويسنده , , B، نويسنده ,
Issue Information :
روزنامه با شماره پیاپی سال 1996
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Abstract :
The Korteweg–de Vries equation (KdV)[formula]is a completely integrable Hamiltonian system of infinite dimension with phase space the Sobolev spaceHN(S1; R), (N⩾1), Hamiltonian H(q):=∫S1(12(∂xq(x))2+q(x)3) dx, and Poisson structure ∂/∂x. The functionq≡0 is an elliptic fixed point. We prove that for anyN⩾1, the Korteweg–de Vries equation (and thus the entire KdV-hierarchy) admits globally defined real analytic action-angle variables. As a consequence it follows that in a neighborhood ofq≡0 inH1(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 vectorfield of KdV.
Journal title :
Journal of Functional Analysis
Journal title :
Journal of Functional Analysis
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