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portada kdv & kam
Type
Physical Book
Author
Publisher
Year
2010
Language
English
Pages
279
Format
Paperback
Dimensions
23.4x15.6x1.6 cm
Weight
0.42 kg.
ISBN
3642056946
ISBN13
9783642056949
Categories

kdv & kam

Thomas Kappeler (Author) · Jürgen Pöschel (Author) · Springer · Paperback

kdv & kam - Kappeler, Thomas ; Pöschel, Jürgen

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Synopsis "kdv & kam"

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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