Fibration of the phase space for the Korteweg-de Vries equation

Thomas Kappeler

Annales de l'institut Fourier (1991)

  • Volume: 41, Issue: 3, page 539-575
  • ISSN: 0373-0956

Abstract

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In this article we prove that the fibration of L 2 ( S 1 ) by potentials which are isospectral for the 1-dimensional periodic Schrödinger equation, is trivial. This result can be applied, in particular, to N -gap solutions of the Korteweg-de Vries equation (KdV) on the circle: one shows that KdV, a completely integrable Hamiltonian system, has global action-angle variables.

How to cite

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Kappeler, Thomas. "Fibration of the phase space for the Korteweg-de Vries equation." Annales de l'institut Fourier 41.3 (1991): 539-575. <http://eudml.org/doc/74929>.

@article{Kappeler1991,
abstract = {In this article we prove that the fibration of $L^ 2(S^ 1)$ by potentials which are isospectral for the 1-dimensional periodic Schrödinger equation, is trivial. This result can be applied, in particular, to $N$-gap solutions of the Korteweg-de Vries equation (KdV) on the circle: one shows that KdV, a completely integrable Hamiltonian system, has global action-angle variables.},
author = {Kappeler, Thomas},
journal = {Annales de l'institut Fourier},
keywords = {isospectral potentials; Schrödinger equation; Korteweg-de Vries equation; completely integrable Hamiltonian system; global action-angle variables},
language = {eng},
number = {3},
pages = {539-575},
publisher = {Association des Annales de l'Institut Fourier},
title = {Fibration of the phase space for the Korteweg-de Vries equation},
url = {http://eudml.org/doc/74929},
volume = {41},
year = {1991},
}

TY - JOUR
AU - Kappeler, Thomas
TI - Fibration of the phase space for the Korteweg-de Vries equation
JO - Annales de l'institut Fourier
PY - 1991
PB - Association des Annales de l'Institut Fourier
VL - 41
IS - 3
SP - 539
EP - 575
AB - In this article we prove that the fibration of $L^ 2(S^ 1)$ by potentials which are isospectral for the 1-dimensional periodic Schrödinger equation, is trivial. This result can be applied, in particular, to $N$-gap solutions of the Korteweg-de Vries equation (KdV) on the circle: one shows that KdV, a completely integrable Hamiltonian system, has global action-angle variables.
LA - eng
KW - isospectral potentials; Schrödinger equation; Korteweg-de Vries equation; completely integrable Hamiltonian system; global action-angle variables
UR - http://eudml.org/doc/74929
ER -

References

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  4. [GT1] J. GARNETT, E. TRUBOWITZ, Gaps and bands of one dimensional periodic Schrödinger operators, Comm. Math. Helv., 59 (1984), 258-312. Zbl0554.34013MR85i:34004
  5. [GT2] J. GARNETT, E. TRUBOWITZ, Gaps and bands of one dimensional periodic Schrödinger operators II, Comm. Math. Helv., 62 (1987), 18-37. Zbl0649.34034MR88g:34028
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  7. [Ka] T. KATO, Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, 1976. Zbl0342.47009MR53 #11389
  8. [Kp] T. KAPPELER, On the periodic spectrum of the 1-dimensional Schrödinger operator, Comm. Math. Helv., 65 (1990), 1-3. Zbl0703.34085MR91a:34059
  9. [Ma] V.A. MARCENKO, Sturm Liouville Operators and Applications, Birkäuser, Basel, 1986. Zbl0592.34011
  10. [MM] H.P. MCKEAN, P. VAN MOERBEKE, The spectrum of Hill's equation, Inv. Math., 30 (1975), 217-274. Zbl0319.34024MR53 #936
  11. [MW] W. MAGNUS, W. WINKLER, Hill's Equation, Wiley-Interscience, New York, 1986. 
  12. [MT] H.P. MICKEAN, E. TRUBOWITZ, Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points, CPAM, 24 (1976), 143-226. Zbl0339.34024MR55 #761
  13. [PS] G. POLYA, G. SZEGÖ, Aufgaben und Lehrsätze aus der Analysis, vol. 2, 3rd ed., Grundlehren, Bd 20, Springer-Verlag, New York, 1964. Zbl0122.29704
  14. [PT] J. PÖSCHEL, E. TRUBOWITZ, Inverse Spectral Theory, Academic Press, 1987. Zbl0623.34001

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