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arXiv · 2209.08943

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

Andrey Dymov
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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 $d$.

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Andrey Dymov. 2022-09-19. Asymptotic expansions for a class of singular integrals emerging in non-linear wave systems. https://doi.org/10.1134/s0040577923020010

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