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.
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.
