KAM-persistence of finite-gap solutions

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