Journal of Mathematical Sciences

, Volume 91, Issue 2, pp 2725–2732 | Cite as

Solutions of the membrane equation concentrated near extremal loops

  • A. S. Golubeva


Formal asymptotic solutions of the equation\(\Delta ^2 u - \frac{{\omega ^4 }}{{c^4 \left( {x,y} \right)}}u = 0\) concentrated in the vicinity of an extremal loop with N vertices are constructed by applying the complex version of the ray method. Bibliography: 5 titles.


Gaussian Beam Helmholtz Equation Homogeneous Problem Monodromy Operator Floquet Exponent 
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  1. 1.
    I. M. Babakov,Theory of Oscillations [in Russian], Moscow (1968).Google Scholar
  2. 2.
    V. M. Babie and V. S. Buldyrev,Short-Wavelength Diffraction Theory, Springer-Verlag, Berlin (1991).Google Scholar
  3. 3.
    M. M. Popov, “Asymptotics of certain subsequences of eigenvalues of boundary-value problems for the Helmholtz equation in the multidimensional case,”Dokl. Akad. Nauk SSSR,184, 1076–1097 (1969).MATHGoogle Scholar
  4. 4.
    V. E. Nomofilov, “Asymptotic solutions of systems of second-order equations concentrated near a ray,”Zap. Nauchn. Semin. LOMI,104, 170–179 (1981).MATHMathSciNetGoogle Scholar
  5. 5.
    P. K. Rashevskii,Lectures on Differential Geometry [in Russian], Gostekhizdat (1956).Google Scholar

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© Plenum Publishing Corporation 1998

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  • A. S. Golubeva

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