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.
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.
G. A. Omel'yanov, D. A. Kulagin, “Asymptotics of Kink–Kink Interaction for Sine-Gordon Type Equations”, Mat. Zametki, 75:4 (2004), 603–607; Math. Notes, 75:4 (2004), 563–567
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
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; Math. Notes, 66:2 (1999), 181–189
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; Math. Notes, 63:1 (1998), 137–139
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; Differ. Equ., 31:3 (1995), 446–454
V. P. Maslov, G. A. Omel'yanov, “The turbulent dynamo problem”, Mat. Zametki, 58:6 (1995), 936–939; Math. Notes, 58:6 (1995), 1352–1355
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; Math. Notes, 57:5 (1995), 559–561
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; Sb. Math., 186:12 (1995), 1753–1771
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; Theoret. and Math. Phys., 98:2 (1994), 202–211
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; Math. Notes, 53:6 (1993), 611–621
G. A. Omel'yanov, “Interaction of waves of different scales in gas dynamics”, Mat. Zametki, 53:1 (1993), 148–151; Math. Notes, 53:1 (1993), 107–109
V. P. Maslov, G. A. Omel'yanov, “Rapidly oscillating asymptotic solution of magnetohydrodynamic equations in the Tokamak approximation”, TMF, 92:2 (1992), 269–292; Theoret. and Math. Phys., 92:2 (1992), 879–895
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
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
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; Math. USSR-Sb., 50:1 (1985), 191–212
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
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; Russian Math. Surveys, 36:3 (1981), 73–149