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Construction of Quasi-periodic solutions with the same Gevrey index as nonlinear terms in Multi-Dimensional NLS

Zuhong You, Xiaoping Yuan

TL;DR

The paper proves the persistence of Gevrey-smooth quasi-periodic solutions for a multi-dimensional NLS with Gevrey nonlinearity of index $\alpha>1$, using the Craig-Wayne-Bourgain (CWB) framework. A Lyapunov-Schmidt reduction separates finite resonant and infinite nonresonant modes, and a Newton-type iteration (with careful Gevrey-function estimates) constructs approximate solutions $q_r$ with finite spectral support that converge to a genuine quasi-periodic solution. Key innovations include direct use of Gevrey norms (no analytic-approximation step), polynomial truncations to enable semi-algebraic measure estimates, and a sharpened coupling lemma that yields arbitrarily small decay losses in the resonant analysis, allowing arbitrary $\alpha>1$. The result yields a Cantor set of parameters with full-measure convergence as $\varepsilon\to0$, and the persisted hull lies in the Gevrey class $G^{\alpha,L_0/2}$ with the drift frequency $\lambda'$ staying close to the linear frequencies. This advances the CWB approach to Gevrey nonlinearities in higher dimensions and broadens the scope of quasi-periodic solutions in nonlinear dispersive PDEs.

Abstract

We investigate the persistency of quasi-periodic solutions to multi-dimensional nonlinear Schrödinger equations (NLS) involving Gevrey smooth nonlinearity with an arbitrary Gevrey index $α>1$. By applying the Craig-Wayne-Bourgain (CWB) method, we establish the existence of quasi-periodic solutions that are Gevrey smooth with the same Gevrey index as the nonlinearity.

Construction of Quasi-periodic solutions with the same Gevrey index as nonlinear terms in Multi-Dimensional NLS

TL;DR

The paper proves the persistence of Gevrey-smooth quasi-periodic solutions for a multi-dimensional NLS with Gevrey nonlinearity of index , using the Craig-Wayne-Bourgain (CWB) framework. A Lyapunov-Schmidt reduction separates finite resonant and infinite nonresonant modes, and a Newton-type iteration (with careful Gevrey-function estimates) constructs approximate solutions with finite spectral support that converge to a genuine quasi-periodic solution. Key innovations include direct use of Gevrey norms (no analytic-approximation step), polynomial truncations to enable semi-algebraic measure estimates, and a sharpened coupling lemma that yields arbitrarily small decay losses in the resonant analysis, allowing arbitrary . The result yields a Cantor set of parameters with full-measure convergence as , and the persisted hull lies in the Gevrey class with the drift frequency staying close to the linear frequencies. This advances the CWB approach to Gevrey nonlinearities in higher dimensions and broadens the scope of quasi-periodic solutions in nonlinear dispersive PDEs.

Abstract

We investigate the persistency of quasi-periodic solutions to multi-dimensional nonlinear Schrödinger equations (NLS) involving Gevrey smooth nonlinearity with an arbitrary Gevrey index . By applying the Craig-Wayne-Bourgain (CWB) method, we establish the existence of quasi-periodic solutions that are Gevrey smooth with the same Gevrey index as the nonlinearity.

Paper Structure

This paper contains 19 sections, 27 theorems, 397 equations.

Table of Contents

  1. Introduction
  2. Gevrey functions and their properties
  3. Lyapounov-Schmidt decomposition
  4. The homological equations and the estimate of the remians
  5. The Iterative Lemma
  6. Estimate of the Green function in first finite steps
  7. Polynomials replacement in each step
  8. Construction of $q_{r}$
  9. The reality of $\lambda'$ and the approximation of $Q$-equations
  10. Measure Estimates
  11. Coupling Lemma
  12. The matrices $T_{N}^{\sigma}$ and its estimation
  13. Estimation of $G_{N_{r+1}}$
  14. Proof of Theorem \ref{['zhudingli']}
  15. A technical proposition about the reality of Fourier coefficients
  16. ...and 4 more sections

Key Result

Theorem 1.1

For arbitrary $\alpha>1$, let $H(q,\bar{q})=f(q\bar{q})$, $f,f',f"\in G^{\alpha,L_0}([-1,1])$$( L_0>0)$. Fix $a_{1},...,a_{b}\in\mathbb{R}_{+}$ and let $I_0=[0,1]^{b}$. For $\varepsilon$ sufficiently small, there exists a Cantor set $I_{0,\varepsilon}\subset I_0$ such that $mes(I_0\setminus I_{0,\va and which implies where $\hat{q}(n,k)$ is the $(n,k)$-Fourier coefficient of $q(\tilde{x},\theta)

Theorems & Definitions (53)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 2.1: Product
  • Proposition 2.2: Derivative
  • Proposition 2.3: Composition
  • Proposition 2.4: Decay of Fourier coefficients
  • proof
  • ...and 43 more