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<p>For the discrete nonlinear quasi-periodic wave equation on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {Z}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>, we establish the long-time stability of small-amplitude solutions. Although the perturbation lacks gauge invariance, we prove, via a partial normal form reduction, that the solutions remain localized for polynomially long times. Our result applies to the entire phase space, without restricting the number of active modes in the initial data, in particular, the initial data need not be finitely supported.</p>

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Long-Time Stability for the Nonlinear Quasi-Periodic Wave Equation on \(\mathbb {Z}^d\)

  • 期刊论文

  • 发表日期：

    2026年6月18日

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  • W.-M. Wang,
  • Zhihan Zhang

摘要

For the discrete nonlinear quasi-periodic wave equation on \(\mathbb {Z}^d\) Z d , we establish the long-time stability of small-amplitude solutions. Although the perturbation lacks gauge invariance, we prove, via a partial normal form reduction, that the solutions remain localized for polynomially long times. Our result applies to the entire phase space, without restricting the number of active modes in the initial data, in particular, the initial data need not be finitely supported.