KdV & KAM

KdV & KAM

EnglishPaperback / softbackPrint on demand
Kappeler Thomas
Springer, Berlin
EAN: 9783642056949
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Detailed information

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.

EAN 9783642056949
ISBN 3642056946
Binding Paperback / softback
Publisher Springer, Berlin
Publication date October 19, 2010
Pages 279
Language English
Dimensions 235 x 155
Country Germany
Readership Professional & Scholarly
Authors Kappeler Thomas; Poschel, Jurgen
Illustrations XIII, 279 p.
Edition Softcover reprint of hardcover 1st ed. 2003
Series Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
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