Asymptotic expansions for a class of singular integrals emerging in non-linear wave systems

TL;DR

This paper derives asymptotic expansions for a class of singular integrals relevant to non-linear wave systems, generalizing previous results and aiding in stochastic model analysis.

Contribution

It provides a generalized asymptotic expansion for integrals with singularities, extending prior work to broader classes of functions and critical points.

Findings

Derived asymptotic expansions as   0 for integrals with singular denominators.

Extended previous quadratic form results to more general functions .

Facilitated analysis of stochastic models in non-linear wave systems.

Abstract

We find asymptotical expansions as $\nu \to 0$ for integrals of the form $\int_{\mathbb{R}^d} F(x) / \big(\omega(x)^2 + \nu^2\big)\, dx$, where sufficiently smooth functions $F$ and $\omega$ satisfy natural assumptions for their behaviour at infinity and all critical points of the function $\omega$ from the set $\{\omega(x) = 0\}$ 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 $\omega$ is a non-degenerate quadratic form of the signature $(d/2,d/2)$ with even $d$.