Research Interests:

  • PDE Analysis: spectral stability problems in Hamiltonian systems, evolution equations.
  • Applied Operator Theory: spectral theory of J-self-adjoint operators.
  • Numerical Analysis: parallel algorithms for eigenvalue distributions of sparse matrices (MPI)
  • Sturm-Liouville problems with spectral parameter dependent boundary conditions.

Brief summary of research projects:

2007 – 2008

  Research project 1: ”On the Nature of Ill-Posedness of Forward-Backward Heat Equation”

Application: dynamic of a thin film of liquid which is entrained on the inside of a rotating cylinder for example coating of fluorescent light bulbs or distribution of liquid thermosetting plastic in the rotating mould.

Analytical results: factorization of the indefinite convection-diffusion operator was obtained in terms of first order differential operators (research in progress).

Collaboration: I. Karabash (University of Calgary, Canada), S.G. Pyatkov (Sobolev Institute of Mathematics, Novosibirsk, Russia), V. Strauss (Universidad Simón Bolívar, Venezuela).

  Research project 2: “Eigenvalue Distributions of Quadratic Matrix Pencils with Symmetries”

Application: stability of vortices in multi-dimensional discrete lattices.

Analytical results: upper bound was found for the number of spectrally unstable eigenvalues. (research in progress).

Collaboration: D. Pelinovsky (McMaster University, Canada). .

2003 – 2007

  Research project: “Spectral Stability of Solitary Waves in Dynamical Systems” (Doctoral Thesis, McMaster University)

Application: dynamic of the magneto-acoustic waves in plasma and capillary-gravity water waves, stability of gap solitons in trapped Bose-Einstein condensates, stability of vacuum.

Analytical results: Criteria for stability and instability of solitons were derived in terms of dimensions of sign-definite invariant subspaces using Pontryagin space decomposition method. By one of these criteria it was proved that embedded eigenvalues of negative Krein signature are structurally stable in a linearized fifth – order KdV equation.

Numerical results: the numerical method was constructed to find two-pulse solutions for the fifth-order KdV equation; the MPI (parallel programming library) was used for the implementation of the code

Collaboration: T. Azizov (Voronezh State University, Russia), A. Comech
(Texas A&M University, US), Mason A. Porter (California Institute of Technology)

Advisor: Dr. D. Pelinovsky (McMaster University, Canada).

1994 – 2002

  Research project: “Generalized Resolvent Method and Sturm-Liouville Problem with Spectral Parameter Dependent Boundary Conditions”.

Application: transverse vibrations of rotating beam with tip mass and vibrations of string with set of discrete loads.

Analytical results: the asymptotic of eigenvalues was obtained for different type of Sturm-Liouville problems where the familiar Dirichlet-Neumann boundary conditions were replaced by boundary conditions that depend on the spectral parameter; the inverse problem was solved for the Nevanlinna type boundary conditions

Advisor: Dr. A. V. Strauss (Ulyanovsk Teacher's Training State University, Russia).

1990 – 1993

  Research project: “Boundary Operators and Entire Extensions of Symmetric Operators”.

Analytical results: entire extension of a second-order differentiation operator was constructed using the boundary operator method.

Advisor: Dr. A. V. Strauss (Ulyanovsk Teacher's Training State University, Russia)

1988 – 1990

  Research project: “Automated Construction of Different Sectional Views for a Three-dimensional Model of the Russian Space Station ‘Mir’“. (Master Thesis, Moscow Institute of Physics and Technology)

Numerical results: the numerical algorithm of calculating the area of the shadow of the complicated three-dimensional object was constructed and implemented as a Pascal programming language code; the surfaces of objects (space modules of the station “Mir”) were approximated using triangulation; the dynamical tree – structure was used for a database of the approximated objects.

Advisor: Dr. M. Komarov (M.V.Keldysh Institute of Applied Mathematics, Russia)