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Periodic Response Solutions to Multi-Dimensional Nonlinear Schrödinger equation with unbounded perturbation

Zuhong You, Xiaoping Yuan

TL;DR

This work addresses the problem of obtaining time-periodic solutions to the multi-dimensional nonlinear Schrödinger equation with unbounded fractional-derivative perturbations by applying the Craig–Wayne–Bourgain (CWB) method. The authors construct an initial approximate solution and iteratively correct it via a Newton scheme, carefully controlling the associated linearized (Green) operators on shrinking Cantor-like parameter sets. Central to the analysis are sharp Green-function estimates, achieved through a multi-scale scheme and coupling lemmas, which ensure exponential off-diagonal decay and bounded inverse norms. The results yield a Gevrey-smooth, $\frac{2\pi}{\omega}$-periodic solution for most frequencies $\omega$ in $[1,2]$ with precise quantitative smallness of the residual and Fourier coefficients, providing a step toward understanding unbounded perturbations in higher-dimensional NLS contexts.

Abstract

By applying the Craig-Wayne-Bourgain (CWB) method, we establish the existence of periodic response solutions to multi-dimensional nonlinear Schrödinger equations (NLS) with unbounded perturbation.

Periodic Response Solutions to Multi-Dimensional Nonlinear Schrödinger equation with unbounded perturbation

TL;DR

This work addresses the problem of obtaining time-periodic solutions to the multi-dimensional nonlinear Schrödinger equation with unbounded fractional-derivative perturbations by applying the Craig–Wayne–Bourgain (CWB) method. The authors construct an initial approximate solution and iteratively correct it via a Newton scheme, carefully controlling the associated linearized (Green) operators on shrinking Cantor-like parameter sets. Central to the analysis are sharp Green-function estimates, achieved through a multi-scale scheme and coupling lemmas, which ensure exponential off-diagonal decay and bounded inverse norms. The results yield a Gevrey-smooth, -periodic solution for most frequencies in with precise quantitative smallness of the residual and Fourier coefficients, providing a step toward understanding unbounded perturbations in higher-dimensional NLS contexts.

Abstract

By applying the Craig-Wayne-Bourgain (CWB) method, we establish the existence of periodic response solutions to multi-dimensional nonlinear Schrödinger equations (NLS) with unbounded perturbation.

Paper Structure

This paper contains 5 sections, 7 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Initial approximate solution
  3. The Newton iteration scheme
  4. Estimate of Green function
  5. Coupling Lemma

Key Result

Theorem 1.1

Assume $\alpha<\frac{1}{30(d(2d+2)^{d+2}+2)}$, $P(x,\theta): \mathbb{T}^{d}\times \mathbb{T}\to \mathbb{R}$ is Gevrey smooth, $c$ is small enough and $\lVert P \rVert_{c}<1$. Then, for $\varepsilon>0$ small enough, there exists a set $I_{\varepsilon}\subset [1,2]$ with such that for any $\omega\in [1,2]\setminus I_{\varepsilon}$, there exists a Gevrey smooth function which solves (peiodic force

Theorems & Definitions (15)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • ...and 5 more