Geodesic
Asymptotic expansions for a~class of singular integrals emerging in nonlinear wave systems
Teoretičeskaâ i matematičeskaâ fizika, Tome 214 (2023) no. 2, pp. 179-197

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We find asymptotic expansions as $\nu\to 0$ for integrals of the form $\int_{\mathbb{R}^d}F(x)/(\omega^2(x)+\nu^2)\,dx$, where sufficiently smooth functions $F$ and $\omega$ satisfy natural assumptions on their behavior at infinity and all critical points of $\omega$ in the set $\{\omega(x)=0\}$ 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 $\omega$ is a nondegenerate quadratic form of signature $(d/2,d/2)$ with even $d$.
Keywords: singular integral, asymptotic analysis, wave turbulence, nonlinear waves system. 
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     author = {A. V. Dymov},
     title = {Asymptotic expansions for a~class of singular integrals emerging in nonlinear wave systems},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {179--197},
     publisher = {mathdoc},
     volume = {214},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2023_214_2_a0/}
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A. V. Dymov. Asymptotic expansions for a~class of singular integrals emerging in nonlinear wave systems. Teoretičeskaâ i matematičeskaâ fizika, Tome 214 (2023) no. 2, pp. 179-197. http://geodesic.mathdoc.fr/item/TMF_2023_214_2_a0/