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Mat. Sb. (N.S.), 1981, Volume 115(157), Number 3(7), Pages 478–492 (Mi msb2408)  

A scattering problem in laminar media

A. L. Piatnitski


Abstract: The scattering problem in a laminar medium
$$ \Delta u(x)+k^2q(x_1,…,x_n,x_1/\varepsilon)u(x)=0 $$
with a radiation condition at infinity is considered. The potential $q(x,y)$ is periodic in the variable $y$. Here $k$ is a large parameter, and $\varepsilon$ is a small parameter with $k\sim\varepsilon^{-\alpha}$, $\alpha>1$.
In this paper a formal asymptotic expansion of the solution of this problem is found. To construct it an operator analogous to the canonical Maslov operator is used which acts on a certain Lagrangian manifold not depending on $\varepsilon$. An analogous problem for the Schrödinger equation in a laminar medium is solved.
Bibliography: 10 titles.

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English version:
Mathematics of the USSR-Sbornik, 1982, 43:3, 427–441

Bibliographic databases:

UDC: 517.9
MSC: Primary 35P25, 35C10; Secondary 35J05
Received: 22.05.1979

Citation: A. L. Piatnitski, “A scattering problem in laminar media”, Mat. Sb. (N.S.), 115(157):3(7) (1981), 478–492; Math. USSR-Sb., 43:3 (1982), 427–441

Citation in format AMSBIB
\Bibitem{Pia81}
\by A.~L.~Piatnitski
\paper A~scattering problem in laminar media
\jour Mat. Sb. (N.S.)
\yr 1981
\vol 115(157)
\issue 3(7)
\pages 478--492
\mathnet{http://mi.mathnet.ru/msb2408}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=628222}
\zmath{https://zbmath.org/?q=an:0501.35069|0468.35071}
\transl
\jour Math. USSR-Sb.
\yr 1982
\vol 43
\issue 3
\pages 427--441
\crossref{https://doi.org/10.1070/SM1982v043n03ABEH002573}


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