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Localized state for nonlinear disordered stark model

Shengqing Hu, Yingte Sun

Abstract

In this paper, we consider the following nonlinear disordered Stark model: $${\bf i}\partial_tu_n+δ(u_{n+1}+u_{n-1})+nu_n+v_nu_n+ε|u_n|^{2}u_n=0,\quad n\in\mathbb{Z}.$$ By employing the diagonalization of the associated linear operators and the KAM theory for nonlinear Hamiltonian systems, we establish that for parameters $δ$ and $\varepsilon$ in a reasonable range, and for most realization of random variables $v=\{v_n\}_{n \in \mathbb{Z}}$, there exist time quasi-periodic and spatially localized states that exhibit arbitrary power-law spatial decay.

Localized state for nonlinear disordered stark model

Abstract

In this paper, we consider the following nonlinear disordered Stark model: By employing the diagonalization of the associated linear operators and the KAM theory for nonlinear Hamiltonian systems, we establish that for parameters and in a reasonable range, and for most realization of random variables , there exist time quasi-periodic and spatially localized states that exhibit arbitrary power-law spatial decay.
Paper Structure (17 sections, 23 theorems, 296 equations)

This paper contains 17 sections, 23 theorems, 296 equations.

Table of Contents

  1. Introduction and main results
  2. Main result
  3. New ingredient of proof
  4. Organization of the paper
  5. The eigenvalue variation
  6. The Hamiltonian structure
  7. Localized state for nonlinear disorder stark model
  8. Functional setting
  9. The abstract KAM theorem
  10. The proof of Theorem \ref{['thm1.1']}
  11. Proof of Theorem \ref{['main theorem']}
  12. The derivation of homological equation
  13. The solvability of homological equation \ref{['huhomo']}
  14. The solution of homological equation
  15. Estimate for the new Hamiltonian
  16. ...and 2 more sections

Key Result

Theorem 1.1

Consider one-dimensional nonlinear disordered stark model eq1.1. Fix $n_k \in \mathbb{Z}, k=1,\cdots, b$ and $d>0$. Let $\mathcal{J}=\{n_k\}^{b}_{k=1} \in \mathbb{Z}$ and $\mathcal{V}=\{v_{\alpha}\}_{\alpha \in \mathcal{J}} \in \mathbb{R}^b$. Then for $0<\delta< \delta_0=O(\frac{1}{b})$, there exits such that if $\mathcal{V} \in \mathcal{O}_\epsilon$, then equation eq1.1 has solution $u(x,t)$ of t

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 35 more