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KAM-persistence of finite-gap solutions

  • Sergei B. KuksinEmail author
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1784)

Contents.

  • Introduction

  • 1 Some analysis in Hilbert spaces and scales
    • 1.1 Smooth and analytic maps

    • 1.2 Scales of Hilbert spaces and interpolation

    • 1.3 Differential forms

  • 2 Symplectic structures and Hamiltonian equations
    • 2.1 Basic definitions

    • 2.2 Symplectic transformations

    • 2.3 Darboux lemmas

    • Appendix. Time-quasiperiodic solutions

  • 3 Lax-integrable Hamiltonian equations and their integrable subsystems
    • 3.1 Examples of Hamiltonian PDEs

    • 3.2 Lax-integrable equations

    • 3.3 Integrable subsystems

  • 4 Finite-gap manifolds and theta-formulas
    • 4.1 Finite-gap manifolds for the KdV equation

    • 4.2 The Its-Matveev theta-formulas

    • 4.3 Higher equations from the KdV hierarchy

    • 4.4 Sine-Gordon equation under Dirichlet boundary conditions

  • 5 Linearised equations and their Floquet solutions
    • 5.1 The linearised equation

    • 5.2 Floquet solutions

    • 5.3 Complete systems of Floquet solutions

    • 5.4 Lower-dimensional invariant tori of finite-dimensional systems and Floquet’s theorem

  • 6 Linearised Lax-integrable equations
    • 6.1 Abstract situation

    • 6.2 Linearised KdV equation

    • 6.3 Higher KdV-equations

    • 6.4 Linearised SG equation

  • 7 Normal form
    • 7.1 A normal form theorem

    • 7.2 Examples

  • 8 The KAM theorem
    • 8.1 The main theorem and related results

    • 8.2 Reduction to a parameter-depending case

    • 8.3 A KAM-theorem for parameter-depending equations

    • 8.4 Completion of the Main Theorem’s proof (Step 4)

  • 9 Examples
    • 9.1 Perturbed KdV equation

    • 9.2 Higher KdV equations

    • 9.3 Perturbed SG equation

    • 9.4 KAM-persistence of lower-dimensional invariant tori of nonlinear finite-dimensional systems

  • References

Mathematics Subject Classification (2000):

37C55 37F25 37F50 37J40 37K55 47B39 34L40 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  1. 1.Department of MathematicsHeriot-Watt UniversityEdingurghUnited Kingdom

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