论文标题
在非线性波系统中出现的一类奇异积分的渐近扩展
Asymptotic expansions for a class of singular integrals emerging in non-linear wave systems
论文作者
论文摘要
We find asymptotical expansions as $ν\to 0$ for integrals of the form $\int_{\mathbb{R}^d} F(x) / \big(ω(x)^2 + ν^2\big)\, dx$, where sufficiently smooth functions $F$ and $ω$ satisfy natural assumptions for their behaviour at infinity and all critical points of the function $ω$ from the设置$ \ {ω（x）= 0 \} $是非脱位的。在分析非线性波系统的随机模型时，这些渐近学起着至关重要的作用。我们的结果概括了[S.库金，拉斯。 J. Math。 phys.'2017]在特定情况下，$ω$是签名$（d/2，d/2）$的特定情况下发现了类似的渐近学，甚至是$ d $。
We find asymptotical expansions as $ν\to 0$ for integrals of the form $\int_{\mathbb{R}^d} F(x) / \big(ω(x)^2 + ν^2\big)\, dx$, where sufficiently smooth functions $F$ and $ω$ satisfy natural assumptions for their behaviour at infinity and all critical points of the function $ω$ from the set $\{ω(x) = 0\}$ are non-degenerate. These asymptotics play a crucial role when analysing stochastic models for non-linear waves systems. Our result generalizes that of [S. Kuksin, Russ. J. Math. Phys.'2017] where a similar asymptotics was found in a particular case when $ω$ is a non-degenerate quadratic form of the signature $(d/2,d/2)$ with even $d$.
