Asymptotic expansions for a class of singular integrals emerging in nonlinear wave systems
Abstract
We find asymptotic expansions as $ν\to 0$ for integrals of the form $\int_{\mathbb{R}^d}F(x)/(ω^2(x)+ν^2) dx$, where sufficiently smooth functions $F$ and $ω$ satisfy natural assumptions on their behavior at infinity and all critical points of $ω$ in the set $\{ω(x)=0\}$ are nondegenerate. These asymptotic expansions play a crucial role in analyzing stochastic models for nonlinear waves systems. We generalize a result of Kuksin that a similar asymptotic expansion occurs in a particular case where $ω$ is a nondegenerate quadratic form of signature $(d/2,d/2)$ with even $d$.
 Publication:

Theoretical and Mathematical Physics
 Pub Date:
 February 2023
 DOI:
 10.1134/S0040577923020010
 arXiv:
 arXiv:2209.08943
 Bibcode:
 2023TMP...214..153D
 Keywords:

 singular integral;
 asymptotic analysis;
 wave turbulence;
 nonlinear waves system;
 Mathematical Physics
 EPrint:
 18 pages