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Quasi-periodic Dynamics for Multi-dimensional Quasi-linear Schrödinger Equations via Resonant Mode Control

Zuhong You, Xiaoping Yuan

Abstract

This paper focuses on the problem of quasi-periodic solutions for multi-dimensional quasi-linear Schrödinger equation. To address the challenge of unbounded perturbations caused by quasi-linear terms in the equation, we define the resonant mode set $\mathcal{K}$ to control nonlinear resonant effects. Combining KAM (Kolmogorov-Arnold-Moser) ( or Nash-Moser ) theory and Fourier analysis methods, we prove that there are plenty of quasi-periodic solutions of the equation. We also present the Fourier expansion form of the solutions and the estimation of frequency shifts.

Quasi-periodic Dynamics for Multi-dimensional Quasi-linear Schrödinger Equations via Resonant Mode Control

Abstract

This paper focuses on the problem of quasi-periodic solutions for multi-dimensional quasi-linear Schrödinger equation. To address the challenge of unbounded perturbations caused by quasi-linear terms in the equation, we define the resonant mode set to control nonlinear resonant effects. Combining KAM (Kolmogorov-Arnold-Moser) ( or Nash-Moser ) theory and Fourier analysis methods, we prove that there are plenty of quasi-periodic solutions of the equation. We also present the Fourier expansion form of the solutions and the estimation of frequency shifts.
Paper Structure (9 sections, 11 theorems, 115 equations)

This paper contains 9 sections, 11 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Lyapunov–Schmidt Reduction
  3. The Range equation
  4. The bifurcation equations
  5. The structure of the resonant set $\mathcal{R}$
  6. Solving the bifurcation equations
  7. Q-I equations
  8. Q-II equations
  9. Proof of Theorem \ref{['dingli1.4']}

Key Result

Theorem 1.1

Suppose that $\rho$ satisfies Diophantine conditions for some $\gamma>0$ and $\tau>1$. Select linearly independent $n_{1},...,n_{b}\in \mathbb{Z}^{d}$ ($d\ge b$) which satisfies that If $\varepsilon$ is sufficiently small, for all $(a_{1},...,a_{b})\in[\varepsilon,2\varepsilon]^{b}$, the quasi-periodic (periodic) solution feiraojie persists with frequency

Theorems & Definitions (21)

  • Definition 1.1: Resonant Modes
  • Theorem 1.1
  • Corollary 1.2
  • Definition 1.2
  • Remark 1.1
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • ...and 11 more