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Sub-Power Law Decay of the Wave Packet Maximum in Disordered Anharmonic Chains

Wojciech De Roeck, Lydia Giacomin, Amirali Hannani, Francois Huveneers

TL;DR

The paper addresses whether nonlinearity can destroy Anderson localization in 1D disordered KG and DNLS chains. It develops an approximate fluctuation-dissipation decomposition of energy current and a high-order perturbative expansion around the harmonic dynamics to control long-time energy transport, complemented by probabilistic resonance bounds. The main result is that the maximum site energy $M(t)$ decays slower than any power of $t$, with a random time ${\mathsf t}^*$ having finite moments such that $M(t) \ge e^{-2(\ln t)^{3/4}}$ for $t \ge {\mathsf t}^*$; this sub-power-law decay holds almost surely for arbitrary finite energy. The approach provides a rigorous framework connecting dynamical bounds, resonance probabilities, and perturbative constructions, and extends to the DNLS chain, highlighting ultra-slow spreading in disordered nonlinear lattices and offering a bridge between numerical observations and mathematical theory.

Abstract

We show that the peak of an initially localized wave packet in one-dimensional nonlinear disordered chains decays more slowly than any power law of time. The systems under investigation are Klein-Gordon and nonlinear disordered Schrödinger-type chains, characterized by a harmonic onsite disordered potential and quartic nearest-neighbor coupling. Our results apply in the long-time limit, hold almost surely, and are valid for arbitrary finite energy values.

Sub-Power Law Decay of the Wave Packet Maximum in Disordered Anharmonic Chains

TL;DR

The paper addresses whether nonlinearity can destroy Anderson localization in 1D disordered KG and DNLS chains. It develops an approximate fluctuation-dissipation decomposition of energy current and a high-order perturbative expansion around the harmonic dynamics to control long-time energy transport, complemented by probabilistic resonance bounds. The main result is that the maximum site energy decays slower than any power of , with a random time having finite moments such that for ; this sub-power-law decay holds almost surely for arbitrary finite energy. The approach provides a rigorous framework connecting dynamical bounds, resonance probabilities, and perturbative constructions, and extends to the DNLS chain, highlighting ultra-slow spreading in disordered nonlinear lattices and offering a bridge between numerical observations and mathematical theory.

Abstract

We show that the peak of an initially localized wave packet in one-dimensional nonlinear disordered chains decays more slowly than any power law of time. The systems under investigation are Klein-Gordon and nonlinear disordered Schrödinger-type chains, characterized by a harmonic onsite disordered potential and quartic nearest-neighbor coupling. Our results apply in the long-time limit, hold almost surely, and are valid for arbitrary finite energy values.

Paper Structure

This paper contains 17 sections, 9 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments:
  3. Model and Result
  4. A Priori Bounds
  5. Light-Cone-Type Bound
  6. A Priori Lower Bound on $M(t)$
  7. Approximate Fluctuation-Dissipation Decomposition
  8. Diagonalization of the Harmonic Dynamics
  9. Good Behavior of the Perturbative Expansion
  10. Explicit Representations
  11. Probabilistic Bounds
  12. Probability of Resonances
  13. Resonant Points and Resonant Intervals
  14. Concluding the Proof
  15. Improved Lower Bound on $M(t)$
  16. ...and 2 more sections

Key Result

Theorem 1

There exists a random variable ${\mathsf t}^*\ge 1$ with finite moments of all orders such that

Theorems & Definitions (19)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 9 more