AuthorKappeler, Thomas. author
TitleKdV & KAM [electronic resource] / by Thomas Kappeler, Jรผrgen Pรถschel
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2003
Edition 3. Folge
Connect tohttp://dx.doi.org/10.1007/978-3-662-08054-2
Descript XIII, 279 p. online resource

SUMMARY

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


CONTENT

I The Beginning -- II Classical Background -- III Birkhoff Coordinates -- IV Perturbed KdV Equations -- V The KAM Proof -- VI Kuksinโs Lemma -- VII Background Material -- VIII Psi-Functions and Frequencies -- IX Birkhoff Normal Forms -- X Some Technicalities -- References -- Notations


SUBJECT

  1. Popular works
  2. Mathematics
  3. Dynamics
  4. Ergodic theory
  5. Global analysis (Mathematics)
  6. Manifolds (Mathematics)
  7. Partial differential equations
  8. Physics
  9. Education
  10. Popular Science
  11. Popular Science in Education
  12. Global Analysis and Analysis on Manifolds
  13. Mathematics
  14. general
  15. Dynamical Systems and Ergodic Theory
  16. Partial Differential Equations
  17. Mathematical Methods in Physics