Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Long time stability for KAM tori of the derivative nonlinear Schrödinger equation

Shengqing Hu, Huining Xue, Xiaoping Yuan

Abstract

This paper is concerned with the long time stability of KAM tori for a class of derivative nonlinear Schrödinger equations subjected to periodic boundary condition.

Long time stability for KAM tori of the derivative nonlinear Schrödinger equation

Abstract

This paper is concerned with the long time stability of KAM tori for a class of derivative nonlinear Schrödinger equations subjected to periodic boundary condition.
Paper Structure (26 sections, 17 theorems, 671 equations, 2 tables)

This paper contains 26 sections, 17 theorems, 671 equations, 2 tables.

Table of Contents

  1. Introduction and main results
  2. KAM theorem
  3. Preliminary
  4. Hamiltonian formulation of DNLS equation
  5. Some notations
  6. The abstract results
  7. Proof of Theorem \ref{['normaltheorem']}
  8. Derivation of homological equations
  9. Solving the homological equations
  10. The solution of Homological equation \ref{['homological']}
  11. The new normal form
  12. Estimate the new high order term and the perturbation
  13. Iterative and convergence
  14. Proof of Theorem \ref{['normaltheorem']}
  15. Long time stability theorem
  16. ...and 11 more sections

Key Result

Theorem 1.1

Consider equation main equation. Given an integer $n\geq 1$ and a real number $p\geq 1$. If $\epsilon$ is sufficiently small, then there exists a positive measure set ${\Pi}_\eta\in\Pi$ with where $\eta$ is some constant in $(0,1)$. For any $\xi\in {\Pi}_\eta$, the nonlinear Schrödinger equation main equation possesses a linearly stable $n$-dimensional KAM torus $\mathcal{T}_\xi$ in Sobolev space

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 30 more