We study the discrete nonlinear random wave equation <Equation ID="Equ89"> <EquationSource Format="TEX">$ u_{tt} -(\varepsilon \Delta -V) u+\delta |u|^{2 p} u =0\quad (p\in \mathbb {N}^+) $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>u</mi> <mrow> <mi mathvariant="italic">tt</mi> </mrow> </msub> <mo>-</mo> <mrow> <mo stretchy="false">(</mo> <mi>ε</mi> <mi mathvariant="normal">Δ</mi> <mo>-</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>+</mo> <mi>δ</mi> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </msup> <mi>u</mi> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">N</mi> </mrow> <mo>+</mo> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </Equation>on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {Z}^{d} \times [0, \infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>d</mi> </msup> <mo>×</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$0<\varepsilon , \delta \ll 1,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>ε</mi> <mo>,</mo> <mi>δ</mi> <mo>≪</mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Δ</mi> </math></EquationSource> </InlineEquation> is the discrete Laplacian and V is the random potential. We fix the random potential V in a good set, then we use the small amplitudes as parameters to construct quasi-periodic solutions of the nonlinear random wave equation.

Quasi-periodic solutions to nonlinear random wave equations at fixed potential realizations

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发表日期：

2026年5月7日

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Jiansheng Geng,
Yingnan Sun,
W.-M. Wang

摘要

We study the discrete nonlinear random wave equation $ u_{tt} -(\varepsilon \Delta -V) u+\delta |u|^{2 p} u =0\quad (p\in \mathbb {N}^+) $ $u_{tt} - (ε Δ - V) u + δ {| u |}^{2 p} u = 0 (p \in N^{+})$ on $\mathbb {Z}^{d} \times [0, \infty )$ $Z^{d} \times [0, \infty)$ , where $0<\varepsilon , \delta \ll 1,$ $0 < ε, δ ≪ 1,$ $\Delta $ $Δ$ is the discrete Laplacian and V is the random potential. We fix the random potential V in a good set, then we use the small amplitudes as parameters to construct quasi-periodic solutions of the nonlinear random wave equation.