Omel'yanov, Georgiy Aleksandrovich

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Total publications: 21
Scientific articles: 21

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Doctor of physico-mathematical sciences (1993)
Speciality: 01.01.02 (Differential equations, dynamical systems, and optimal control)
Birth date: 03.10.1950
Keywords: nonlinear differential equations; asymptotics solutions; generalized solutions.


Asymptotic solutions of non-linear differential equations of mathematical physics with a small parameter. Non-smooth solutions of non-linear equations. A geometric asymptotics method for obtaining asymptotic solutions of non- linear partial differential equations has been developed (in cooperation with V. P. Maslov). The method allows to obtain solutions with local fast variation (in particular, soliton type and shock wave type solutions) for nonintegrable multidimensional equations. A method for the calculation of rapidly oscillating asymptotic solutions was developed (in cooperation with V. P. Maslov). These solutions describe wave interactions in weak non-linear multidimensional media with small dispersion or viscosity. New model equations were derived, in particular, a generalization of the Kadomtsev–Pogutse equations describing the torus effects, heat transfer effects and generation of longitudinal components of magnetic field and velocity. A weak asymptotics method for the calculation of dynamics and interactions of nonlinear waves for nonintegrable nonlinear PDE is under construction (in cooperation with V. G. Danilov). The interaction of solitary waves for the KdV type equations with small dispersion and merging of free interfaces in the modified Stefan problem were described.


Graduated from Faculty of Apllied Mathematics of Moscow State Institute of Electronics and Mathematics in 1965 (department of Apllied Mathematics). Ph.D. thesis was defended in 1981. D.Sci. thesis was defended in 1993. Full professor since 1996. A list of my works contains more than 90 titles.

Main publications:
  • Maslov V. P. and Omel'yanov G. A. Geometric Asymptotics for Nonlinear PDE. Translations of Mathematical Monographs, A. M. S., 202, 2001.
  • Maslov V. P. and Omel'yanov G. A. Nonlinear evolution of fluctuations in the Tokamak plasma and dynamics of the plasma pinch boundary // Fizika Plasmy, 1995, v. 21, no 8, 684–696. English transl. in Plasma Physics.
  • Omel'yanov G. A., Danilov V. G. and Radkevich E. V. Asymptotic solution of the conserved phase field system in the fast relaxation case // Europ.J.Appl. Math., 1998, v. 9, 1–21.
  • Danilov V. G., Omel'yanov G. A. and Radkevich E. V. Hugoniot–type conditions and weak solutions to the phase field system // Europ.J.Appl. Math., 1999, v. 10, 55–77.
  • Danilov V. G. and Omel'yanov G. A. Calculation of the singularity dynamics for quadratic nonlinear hyperbolic equations. Example: the Hopf equation. In: Nonlinear Theory of Generalized Functions, M. Grosser at all (eds.) // Research Notes in Mathematics, no. 401, Chapman and Hall, London, 1999, 63–74.
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List of publications on ZentralBlatt

Publications in Math-Net.Ru
1. G. A. Omel'yanov, D. A. Kulagin, “Asymptotics of Kink–Kink Interaction for Sine-Gordon Type Equations”, Mat. Zametki, 75:4 (2004),  603–607  mathnet  mathscinet  zmath  elib; Math. Notes, 75:4 (2004), 563–567  isi
2. D. A. Kulagin, G. A. Omel'yanov, N. O. Ordinartseva, “Numerical simulation of unstable processes in phase decomposition problem”, Matem. Mod., 14:2 (2002),  27–38  mathnet  mathscinet  zmath
3. G. A. Omel'yanov, V. V. Trushkov, “Dynamics of a free boundary in a binary medium with variable thermal conductivity”, Mat. Zametki, 66:2 (1999),  231–241  mathnet  mathscinet  zmath; Math. Notes, 66:2 (1999), 181–189  isi
4. G. A. Omel'yanov, V. V. Trushkov, “A geometric correction in the problem on the motion of a free boundary”, Mat. Zametki, 63:1 (1998),  151–153  mathnet  mathscinet  zmath; Math. Notes, 63:1 (1998), 137–139  isi
5. G. A. Omel'yanov, E. V. Radkevich, V. G. Danilov, “The problem of phase transition in a phase field system”, Dokl. Akad. Nauk, 352:6 (1997),  731–734  mathnet  mathscinet  zmath
6. V. G. Danilov, G. A. Omel'yanov, E. V. Radkevich, “Asymptotic behavior of the solution of a phase field system, and a modified Stefan problem”, Differ. Uravn., 31:3 (1995),  483–491  mathnet  mathscinet; Differ. Equ., 31:3 (1995), 446–454
7. V. P. Maslov, G. A. Omel'yanov, “The turbulent dynamo problem”, Mat. Zametki, 58:6 (1995),  936–939  mathnet  mathscinet  zmath; Math. Notes, 58:6 (1995), 1352–1355  isi
8. G. A. Omel'yanov, V. G. Danilov, E. V. Radkevich, “On regularization of initial conditions of the modified Stefan problem”, Mat. Zametki, 57:5 (1995),  793–795  mathnet  mathscinet  zmath; Math. Notes, 57:5 (1995), 559–561  isi
9. V. G. Danilov, G. A. Omel'yanov, E. V. Radkevich, “Justification of asymptotics of solutions of the phase-field equations and a modified Stefan problem”, Mat. Sb., 186:12 (1995),  63–80  mathnet  mathscinet  zmath; Sb. Math., 186:12 (1995), 1753–1771  isi
10. V. P. Maslov, G. A. Omel'yanov, “Turbulent bending of a plasma filament in vacuum”, Dokl. Akad. Nauk, 336:3 (1994),  320–323  mathnet  mathscinet  zmath; Dokl. Math., 39:5 (1994), 308–311
11. V. P. Maslov, G. A. Omel'yanov, “Three-scale expansion of the solution of the magnetohydrodynamic equations and the Reynolds equation for a tokamak”, TMF, 98:2 (1994),  297–311  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 98:2 (1994), 202–211  isi
12. G. A. Omel'yanov, “Existence of a solution to the equations of magnetohydrodynamics with helical symmetry in the tokamak approximation”, Mat. Zametki, 53:6 (1993),  72–88  mathnet  mathscinet  zmath; Math. Notes, 53:6 (1993), 611–621  isi
13. G. A. Omel'yanov, “Interaction of waves of different scales in gas dynamics”, Mat. Zametki, 53:1 (1993),  148–151  mathnet  mathscinet  zmath; Math. Notes, 53:1 (1993), 107–109  isi
14. V. P. Maslov, G. A. Omel'yanov, “On equations of Kadomtsev–Pogutse type for a tokamak and domains of arbitrary symmetry”, Dokl. Akad. Nauk, 326:1 (1992),  83–90  mathnet  mathscinet; Dokl. Math., 37:9 (1992), 480–483
15. V. P. Maslov, G. A. Omel'yanov, “Rapidly oscillating asymptotic solution of magnetohydrodynamic equations in the Tokamak approximation”, TMF, 92:2 (1992),  269–292  mathnet  mathscinet  zmath; Theoret. and Math. Phys., 92:2 (1992), 879–895  isi
16. G. A. Omel'yanov, “Interaction of short waves with nonlinear phases in weakly nonlinear media with small viscosity”, Mat. Zametki, 48:5 (1990),  150–153  mathnet  mathscinet  zmath
17. G. A. Omel'yanov, “Interaction of short waves in the barotropic atmosphere”, Dokl. Akad. Nauk SSSR, 307:4 (1989),  844–849  mathnet  mathscinet; Dokl. Math., 34:8 (1989), 684–687
18. V. P. Maslov, G. A. Omel'yanov, “Soliton-like asymptotics of internal waves in a stratified fluid with small dispersion”, Differ. Uravn., 21:10 (1985),  1766–1775  mathnet  mathscinet
19. V. P. Maslov, G. A. Omel'yanov, V. A. Tsupin, “Asymptotics of some differential and pseudodifferential equations, and dynamical systems with small dispersion”, Mat. Sb. (N.S.), 122(164):2(10) (1983),  197–219  mathnet  mathscinet  zmath; Math. USSR-Sb., 50:1 (1985), 191–212
20. G. A. Omel'yanov, “Boundary value problems for elliptic systems of nonlinear differential equations with a small parameter”, Differ. Uravn., 18:10 (1982),  1829–1831  mathnet  mathscinet  zmath
21. V. P. Maslov, G. A. Omel'yanov, “Asymptotic soliton-form solutions of equations with small dispersion”, Uspekhi Mat. Nauk, 36:3(219) (1981),  63–126  mathnet  mathscinet  zmath; Russian Math. Surveys, 36:3 (1981), 73–149  isi

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