Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Construction of periodic solutions of multi-dimensional nonlinear wave equations with unbounded perturbation

Huining Xue, Zuhong You, Xiaoping Yuan

TL;DR

The paper advances the persistence theory for nonlinear wave equations by extending the Craig-Wayne-Bourgain framework to multi-dimensional NLW with unbounded, derivative-type perturbations. It combines a lattice formulation with a Lyapunov-Schmidt decomposition into P- and Q-equations, and develops a Newton-type analytic scheme for the P-equations while deriving precise Q-equation amplitudes and frequency corrections. A key technical contribution is a separation lemma for unbounded perturbations under generalized Diophantine conditions, together with finite- and infinite-step measure estimates that yield a Cantor-type set of parameters with full asymptotic measure as the perturbation parameter vanishes. The results provide Gevrey-smooth periodic solutions with explicit frequency shifts $\lambda^{2}=|m_{0}|^{2}+\rho+\tfrac{3}{4}\langle m_{0}\rangle^{\alpha} p_{0}^{2}\varepsilon^{2}+O(\varepsilon^{17/8})$, establishing robustness of periodic responses in higher-dimensional NLW under fractional-derivative perturbations.

Abstract

By applying the Craig-Wayne-Bourgain (CWB) method, we establish the persistence of periodic solutions to multi-dimensional nonlinear wave equations (NLW) with unbounded perturbation.

Construction of periodic solutions of multi-dimensional nonlinear wave equations with unbounded perturbation

TL;DR

The paper advances the persistence theory for nonlinear wave equations by extending the Craig-Wayne-Bourgain framework to multi-dimensional NLW with unbounded, derivative-type perturbations. It combines a lattice formulation with a Lyapunov-Schmidt decomposition into P- and Q-equations, and develops a Newton-type analytic scheme for the P-equations while deriving precise Q-equation amplitudes and frequency corrections. A key technical contribution is a separation lemma for unbounded perturbations under generalized Diophantine conditions, together with finite- and infinite-step measure estimates that yield a Cantor-type set of parameters with full asymptotic measure as the perturbation parameter vanishes. The results provide Gevrey-smooth periodic solutions with explicit frequency shifts , establishing robustness of periodic responses in higher-dimensional NLW under fractional-derivative perturbations.

Abstract

By applying the Craig-Wayne-Bourgain (CWB) method, we establish the persistence of periodic solutions to multi-dimensional nonlinear wave equations (NLW) with unbounded perturbation.

Paper Structure

This paper contains 11 sections, 8 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Lattice Formulation and Lyapounov-Schmidt decomposition
  3. Solving the P-equations: the analytic part
  4. Solving the Q-equations and estimating the measure
  5. Solving the Q-equations
  6. Measure estimate in finite step
  7. The separation lemma
  8. Measure estimate for infinite steps
  9. Some facts about the generalized Diophantine conditions
  10. Proof of Lemma \ref{['fenlixingdinglifornlw']}
  11. Coupling Lemma

Key Result

Theorem 1.1

Fix $m_{0}$ and suppose $\rho$ satisfies where $\gamma$ is a small constant and $\tilde{C}$ is a large number depending on $d$. Let $\alpha$ and $c$ be sufficient small. If $\varepsilon$ is sufficiently small, then there is a Cantor type set $I_{\varepsilon}\in[1,2]$. For $p_{0}\in I_{\varepsilon}$, the periodic solution persists. The persisted periodic solution is of Gevrey smoothness and with

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark
  • Definition 2.1
  • Definition 3.1
  • Proposition 3.2
  • Remark
  • Lemma 4.1
  • Definition 4.2
  • Remark
  • Proposition A.1
  • ...and 8 more