Abstract:
Series of asymptotic solutions of nonlinear elliptic boundary-value problems in compact domains with a spectral parameter contained in the boundary condition are constructed, and the connection of these solutions with the trajectories of classical Hamiltonian systems defined on the boundary of the domains considered is established. The asymptotic solutions indicated are expressed in terms of multidimensional Dirichlet series, and a superposition law is established for them which, as it turns out, does not depend either on the number of independent variables in the original problem or on the form of the nonlinearity.
Citation:
S. Yu. Dobrokhotov, V. P. Maslov, “Multidimensional Dirichlet series in the problem of the asymptotic behavior of spectral series of nonlinear elliptic operators”, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat., 23, VINITI, Moscow, 1983, 137–222; J. Soviet Math., 28:1 (1985), 91–143
\Bibitem{DobMas83}
\by S.~Yu.~Dobrokhotov, V.~P.~Maslov
\paper Multidimensional Dirichlet series in the problem of the asymptotic behavior of spectral series of nonlinear elliptic operators
\serial Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat.
\yr 1983
\vol 23
\pages 137--222
\publ VINITI
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/intd70}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=734315}
\zmath{https://zbmath.org/?q=an:0564.58035}
\transl
\jour J. Soviet Math.
\yr 1985
\vol 28
\issue 1
\pages 91--143
\crossref{https://doi.org/10.1007/BF02104897}
Linking options:
https://www.mathnet.ru/eng/intd70
https://www.mathnet.ru/eng/intd/v23/p137
This publication is cited in the following 6 articles:
Kinetic Boltzmann, Vlasov and Related Equations, 2011, 289
V. V. Belov, O. S. Dobrokhotov, S. Yu. Dobrokhotov, “Isotropic Tori, Complex Germ and Maslov Index, Normal Forms and Quasimodes of Multidimensional Spectral Problems”, Math. Notes, 69:4 (2001), 437–466
A. V. Nesterov, “The asymptotic form of the solution of an elliptic equation with the singularly perturbed boundary condition”, Comput. Math. Math. Phys., 34:11 (1994), 1477–1481
V. V. Belov, S. Yu. Dobrokhotov, “Semiclassical maslov asymptotics with complex phases. I. General approach”, Theoret. and Math. Phys., 92:2 (1992), 843–868
V. G. Danilov, P. Yu. Subochev, “Wave solutions of semilinear parabolic equations”, Theoret. and Math. Phys., 89:1 (1991), 1029–1046
A. R. Its, A. V. Rybin, M. A. Sall', “Exact integration of nonlinear Schrödinger equation”, Theoret. and Math. Phys., 74:1 (1988), 20–32
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