[Paper Review] The Stability of Full Dimensional KAM tori for Nonlinear Schrödinger equation

Hongzi Cong, Jianjun Liu+2 more|arXiv (Cornell University)|May 3, 2017

Quantum chaos and dynamical systems17 references4 citations

TL;DR

This paper establishes the long-time stability of full-dimensional KAM tori for the 1D nonlinear Schrödinger equation with periodic boundary conditions, building on Bourgain's construction of tori with slow decay $ I_n \sim e^{-r|n|^{1/2}} $. Using a Nash-Moser iteration scheme with tame estimates and harmonic analysis, the authors prove that solutions near these tori remain close for exponentially long times, extending stability results from finite to infinite-dimensional Hamiltonian PDEs.

ABSTRACT

In this paper, it is proved that the full dimensional invariant tori obtained by Bourgain [J. Funct. Anal., extbf{229} (2005), no. 1, 62-94.] is stable in a very long time for 1D nonlinear Schrödinger equation with periodic boundary conditions.

Motivation & Objective

• To establish the long-time stability of full-dimensional KAM tori in the 1D nonlinear Schrödinger equation (NLS) with periodic boundary conditions.
• To extend classical KAM theory to infinite-dimensional Hamiltonian PDEs by addressing the challenge of resonant frequency interactions in the presence of infinitely many degrees of freedom.
• To demonstrate that the full-dimensional KAM tori constructed by Bourgain—featuring actions decaying as $ I_n \sim e^{-r|n|^{1/2}} $—are stable over exponentially long timescales.
• To overcome the difficulty of solving the homological equation in the infinite-dimensional setting by developing a refined iterative scheme with tame estimates.

Proposed method

• Applies a Nash-Moser iterative scheme to construct and stabilize full-dimensional KAM tori in the 1D NLS equation with periodic boundary conditions.
• Uses a Birkhoff normal form procedure to eliminate non-resonant terms through successive transformations, reducing the perturbation to a remainder of controlled size.
• Employs tame estimates for the homological equation $ \{N_*, F_s\} + Q_{ss} = Z_{ss} $, ensuring convergence despite the infinite-dimensional setting.
• Implements a recursive scheme where the solution $ F_s $ of the homological equation is bounded via $ \|F_s\|_{\rho_s + \delta} \leq C_1(\delta, \theta, \gamma) \|Q_{ss}\|_{\rho_s} $, leveraging Diophantine frequency properties.
• Controls the growth of remainder terms $ Q_{s+1} $ and $ Z_{s+1} $ using factorial decay and iterative bounds, ensuring convergence of the series.
• Relies on harmonic analysis and semi-algebraic set theory techniques, extending the C-W-B (Craig-Wayne-Bourgain) method to infinite-dimensional systems.

Experimental results

Research questions

• RQ1 Can the full-dimensional KAM tori constructed by Bourgain for the 1D NLS with periodic boundary conditions be shown to be stable over long timescales?
• RQ2 What is the optimal decay rate of actions $ I_n $ on the KAM torus that still allows for long-time stability in the NLS equation?
• RQ3 Is it possible to extend the classical KAM stability results—valid for finite-dimensional systems with $ |t| \lesssim \exp(\epsilon^{-1/2}) $—to infinite-dimensional PDEs like the NLS with full-dimensional tori?
• RQ4 How can one control the growth of remainder terms in an iterative KAM scheme when the number of degrees of freedom is infinite?
• RQ5 Can the homological equation be solved in the infinite-dimensional setting with tame estimates, enabling convergence of the iterative normalization process?

Key findings

• The full-dimensional KAM tori constructed by Bourgain for the 1D NLS with periodic boundary conditions are stable for exponentially long times, specifically $ |t| \lesssim \exp(\epsilon^{-\frac{1}{2}}) $, matching the classical Nekhoroshev-type estimate.
• The actions on the KAM tori decay as $ I_n \sim e^{-r|n|^{1/2}} $, which is significantly slower than the double-exponential decay $ e^{-|n|^S} $ with $ S>1 $ seen in prior constructions.
• The iterative Nash-Moser scheme converges due to the use of tame estimates, which control the growth of derivatives in the solution of the homological equation.
• The remainder terms $ Q_{s+1} $ and $ Z_{s+1} $ are bounded by $ (C(\delta, \theta, \gamma))^{(s-1)(s + n(s-1))} $, ensuring convergence of the normalization process.
• The stability result holds under Diophantine frequency conditions and relies on the implicit function theorem in a tame Fréchet space framework.
• The method successfully handles the infinite-dimensional nature of the NLS equation by combining harmonic analysis, semi-algebraic set theory, and iterative normal form techniques.