Application: Thin film flows occur over a wide range of length scales and are central to numerous areas of engineering, geophysics, and biophysics; these include coating flows, lava flows, dynamics of continental ice sheets, tear-film rupture, and surfactant replacement therapy.

Analytical results: existence of nonnegtaive periodic weak solutions was proved for some reasonable restrictions on initial data.
Collaboration: M. Pugh (University of Toronto), R. Taranets (Institute of Applied Mathematics and Mechanics of NAS of Ukraine)

Research in progress: convergence of nonnegative weak solution to the steady state in Wasserstein metric.
Collaboration: A. Burchard (University of Toronto), B. Stephens (University of Washington)

Research in progress: stability of periodic, solitary, and shock waves in liquid films on vibrating substrates.
Collaboration: E. Benilov (University of Limerick)

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: completness with the absence of the Riesz basis property was proved for the system of eigenfunctions of the indefinite convection-diffusion operator.

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

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

Analytical results: upper bound was found for the number of spectrally unstable eigenvalues.

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

 

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).

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).

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)

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)