We study the nonlinear discrete random Schrödinger equation <Equation ID="Equ1"> <EquationSource Format="TEX">$ i \frac{\partial }{\partial t} u+(\varepsilon _n \Delta +V) u+\delta |u|^{2 p} u =0 \quad \left( p \in \mathbb {N}^{+}\right) $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>i</mi> <mfrac> <mi>∂</mi> <mrow> <mi>∂</mi> <mi>t</mi> </mrow> </mfrac> <mi>u</mi> <mo>+</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mi>n</mi> </msub> <mi mathvariant="normal">Δ</mi> <mo>+</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>+</mo> <mi>δ</mi> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mn>2</mn> <mi>p</mi> </mrow> </msup> <mi>u</mi> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mfenced close=")" open="("> <mi>p</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">N</mi> </mrow> <mo>+</mo> </msup> </mfenced> </mrow> </math></EquationSource> </Equation>and discrete random wave equation <Equation ID="Equ2"> <EquationSource Format="TEX">$ u_{tt}+(\varepsilon _n \Delta +V) u+\delta u^{2 p+1} =0 \quad (p\in \mathbb {N}^+) $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>u</mi> <mrow> <mi mathvariant="italic">tt</mi> </mrow> </msub> <mo>+</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ε</mi> <mi>n</mi> </msub> <mi mathvariant="normal">Δ</mi> <mo>+</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> <mi>u</mi> <mo>+</mo> <mi>δ</mi> <msup> <mi>u</mi> <mrow> <mn>2</mn> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">N</mi> </mrow> <mo>+</mo> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </Equation>on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\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">$0< \delta <\varepsilon \ll 1,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>δ</mi> <mo><</mo> <mi>ε</mi> <mo>≪</mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Δ</mi> </math></EquationSource> </InlineEquation> is the discrete Laplacian and V is the random potential, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\left| \varepsilon _n\right| \le \varepsilon e^{-\varrho |n|}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfenced close="|" open="|"> <msub> <mi>ε</mi> <mi>n</mi> </msub> </mfenced> <mo>≤</mo> <mi>ε</mi> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>ϱ</mi> <mo stretchy="false">|</mo> <mi>n</mi> <mo stretchy="false">|</mo> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ \varrho >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϱ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. We fix the random potential V in a good set and choose the small amplitudes as parameters to prove a KAM theorem for finding linearly stable quasi-periodic solutions of nonlinear random Schrödinger equation and wave equation.

KAM Theorem for Discrete Nonlinear Random Equations at Fixed Potential Realizations

期刊论文
发表日期：

2026年4月25日

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

摘要

We study the nonlinear discrete random Schrödinger equation $ i \frac{\partial }{\partial t} u+(\varepsilon _n \Delta +V) u+\delta |u|^{2 p} u =0 \quad \left( p \in \mathbb {N}^{+}\right) $ $i \frac{\partial}{\partial t} u + (ε_{n} Δ + V) u + δ {| u |}^{2 p} u = 0 (p \in N^{+})$ and discrete random wave equation $ u_{tt}+(\varepsilon _n \Delta +V) u+\delta u^{2 p+1} =0 \quad (p\in \mathbb {N}^+) $ $u_{tt} + (ε_{n} Δ + V) u + δ u^{2 p + 1} = 0 (p \in N^{+})$ on $\mathbb {Z}^{d} \times [0, \infty )$ $Z^{d} \times [0, \infty)$ , where $0< \delta <\varepsilon \ll 1,$ $0 < δ < ε ≪ 1,$ $\Delta $ $Δ$ is the discrete Laplacian and V is the random potential, $\left| \varepsilon _n\right| \le \varepsilon e^{-\varrho |n|}$ $|ε_{n}| \leq ε e^{- ϱ | n |}$ with $ \varrho >0$ $ϱ > 0$ . We fix the random potential V in a good set and choose the small amplitudes as parameters to prove a KAM theorem for finding linearly stable quasi-periodic solutions of nonlinear random Schrödinger equation and wave equation.