Mathematics-Dynamical Systems

KdV & KAM

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Title (user) : KdV & KAMISBN : 3540022341,9783540022343ISSN : 0071-1136DDC : 519.3LCC : QA377 .K363 2003GoogleBook ID : OgY7TSZbKSoCOpenLibrary ID : OL17722571MEdition : 1Series : Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, 45Authors (user) : Thomas Kappeler, Jürgen PöschelAuthors (google) : Thomas Kappeler,Jürgen PöschelPublisher : SpringerLanguage : EnglishPublication Date : 2003File Format : pdfCategories : Education------------------------------------------Description (user) : 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...
Description

Title (user) : KdV & KAM

ISBN : 3540022341,9783540022343

ISSN : 0071-1136

DDC : 519.3

LCC : QA377 .K363 2003

GoogleBook ID : OgY7TSZbKSoC

OpenLibrary ID : OL17722571M

Edition : 1

Series : Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, 45

Authors (user) : Thomas Kappeler, Jürgen Pöschel

Authors (google) : Thomas Kappeler,Jürgen Pöschel

Publisher : Springer

Language : English

Publication Date : 2003

File Format : pdf

Categories : Education


------------------------------------------

Description (user) :

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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Description (google) :
This text treats the Korteweg-de Vries (KdV) equation with periodic boundary conditions. This equation models waves in homogeneous, weakly nonlinear and weakly dispersive media in general. For the first time, these important results are comprehensively covered in book form, authored by internationally renowned experts in the field.

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