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KdV & KAM


Kappeler, T; Pöschel, J (2003). KdV & KAM. Berlin: Springer.

Abstract

In this text the authors consider the Korteweg-de Vries (KdV) equation (ut = - uxxx + 6uux) with periodic boundary conditions. Derived to describe long surface waves in a narrow and shallow channel, this equation in fact models waves in homogeneous, weakly nonlinear and weakly dispersive media in general.

Viewing the KdV equation as an infinite dimensional, and in fact integrable Hamiltonian system, we first construct action-angle coordinates which turn out to be globally defined. They make evident that all solutions of the periodic KdV equation are periodic, quasi-periodic or almost-periodic in time. Also, their construction leads to some new results along the way.

Subsequently, these coordinates allow us to apply a general KAM theorem for a class of integrable Hamiltonian pde's, proving that large families of periodic and quasi-periodic solutions persist under sufficiently small Hamiltonian perturbations.

The pertinent nondegeneracy conditions are verified by calculating the first few Birkhoff normal form terms -- an essentially elementary calculation.

Abstract

In this text the authors consider the Korteweg-de Vries (KdV) equation (ut = - uxxx + 6uux) with periodic boundary conditions. Derived to describe long surface waves in a narrow and shallow channel, this equation in fact models waves in homogeneous, weakly nonlinear and weakly dispersive media in general.

Viewing the KdV equation as an infinite dimensional, and in fact integrable Hamiltonian system, we first construct action-angle coordinates which turn out to be globally defined. They make evident that all solutions of the periodic KdV equation are periodic, quasi-periodic or almost-periodic in time. Also, their construction leads to some new results along the way.

Subsequently, these coordinates allow us to apply a general KAM theorem for a class of integrable Hamiltonian pde's, proving that large families of periodic and quasi-periodic solutions persist under sufficiently small Hamiltonian perturbations.

The pertinent nondegeneracy conditions are verified by calculating the first few Birkhoff normal form terms -- an essentially elementary calculation.

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Additional indexing

Item Type:Monograph
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Integrable Systems - KAM Theory - KdV Equation - Perturbation Theory
Language:English
Date:2003
Deposited On:08 Dec 2009 14:42
Last Modified:06 Dec 2017 20:49
Publisher:Springer
Series Name:Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]
Volume:45
Number of Pages:279
ISBN:3-540-02234-1
Official URL:http://www.springer.com/mathematics/analysis/book/978-3-540-02234-3
Related URLs:http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1032.37001
http://www.ams.org/mathscinet-getitem?mr=1997070
http://www.recherche-portal.ch/primo_library/libweb/action/search.do?fn=search&mode=Advanced&vid=ZAD&vl%28186672378UI0%29=isbn&vl%281UI0%29=contains&vl%28freeText0%29=3-540-02234-1

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