Birkhoff coordinates for kdv on phase sp
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Birkhoff coordinates for kdv on phase sp
Birkhoff coordinates for KdV on phase spaces of distributionsT. Kappeler* c. Mõhr. p. Topalov138453AbstractThe purpose of this paper is to extend the Birkhoff coordinates for kdv on phase sp construct ion of Birkhoff coordinates for the KdV equation from the phase space of square integrable Ỉ-periodic functions with mean value zero to the phase space /7(7’(T) of mean value zero dist ributions from the Sobolev space y/1 (TT) endowed with the symploctic structure (ỡ/ữr)-1. More precisely Birkhoff coordinates for kdv on phase sp , we constmet a globally defined real analytic symplcctoinor-phism ÍỈ :—> I)-12 where I)-1 is a weighted Hilbert spaceof sequences (.Tn. yn)n>i suppliBirkhoff coordinates for kdv on phase sp
ed with the canonical Poisson structure so that the KdV Hamiltonian for potentials in //(*(T) is a function of the actions ((a:2 + y2)/2)„>! alone.1 IBirkhoff coordinates for KdV on phase spaces of distributionsT. Kappeler* c. Mõhr. p. Topalov138453AbstractThe purpose of this paper is to extend the Birkhoff coordinates for kdv on phase sp the Hamiltonian H is real analytic and takes the form«(~) = EA>(^ + »J)/2+"(1-2)j=l’Supported in part by the Swiss National Science Foundation and by the European liesearch Training Network HPRN-CT-1999-00118♦Supported in part by MESC grant MM-1003/001where z — (x\;ự) 6 R" X R", Xf G R, and (he dot Birkhoff coordinates for kdv on phase sp s stand for terms of higher order in z. Then z — 0 is an elliptic equilibrium of (1.1), i.e. at z — 0. ™ _ 0, ™ _ 0 and the system z - Az withBirkhoff coordinates for kdv on phase sp
rifV,< l)af .4 —\-A oyand A — diag(Ai,.... A„),-1.3obtained by linearizing (1.1) al z, — 0 has the property that spec.4 — {±t'A|,..., ±iAn} is purely Birkhoff coordinates for KdV on phase spaces of distributionsT. Kappeler* c. Mõhr. p. Topalov138453AbstractThe purpose of this paper is to extend the Birkhoff coordinates for kdv on phase sp > 2 is a homogeneous polynomial of order k in 4-, x2n -I- y'~, It can be shown see e.g. 1291that for a real analytic HamiltonianZ/ = ^AJ(I’ + s?)/2+..rj=lwith Al,.... An nonresonant (i.e. Xjkj / 0 for any (Al,, A„) e Zn \ {0}) there exists a formal ẹymplectic transformation ‘1’ = i(l+- • • represent Birkhoff coordinates for kdv on phase sp ed by a formal power series such thatH o <1> = N2 + A 4 + • • •is in Birkhoff normal form as a formal, power series. In general,Birkhoff coordinates for kdv on phase sp
t in any neighborhood of (he origin [31].Note (hat a Hamiltonian system with real analytic Hamiltonian given by a convergent power series of the form Birkhoff coordinates for KdV on phase spaces of distributionsT. Kappeler* c. Mõhr. p. Topalov138453AbstractThe purpose of this paper is to extend the Birkhoff coordinates for kdv on phase sp also true. Ư a real analytic Hamiltonian with a nonresonant elliptic equilibrium admits n functionally independent integrals in involution then one can introduce real analytic symplcctic coordinates (x\ y) near the equilibrium so that when expressed ill these new coordinates, H is in Birkhoff norma Birkhoff coordinates for kdv on phase sp l form - see [3-3. 1'2. -311. Wo refer to2coordinates of this type as BirkhoJJ coordinates. The equations of motion in those coordinates arc■j* - UkVkBirkhoff coordinates for kdv on phase sp
, ỳk -(1 < k < n)where u»fc D— are the frequencies. They are easily integrated by quadrature. Such coordinates are also very useful when studying HamiBirkhoff coordinates for KdV on phase spaces of distributionsT. Kappeler* c. Mõhr. p. Topalov138453AbstractThe purpose of this paper is to extend the Birkhoff coordinates for kdv on phase sp is paper is to construct Birkhoff coordinates for rhe Korreweg - de Vries equation (KdV)Birkhoff coordinates for KdV on phase spaces of distributionsT. Kappeler* c. Mõhr. p. Topalov138453AbstractThe purpose of this paper is to extend theGọi ngay
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