Theoretical physics: control in quantum physical models;quantum optics, nanolithography,
focusing of quantum particles, beam-splitters, cooling; turbulence;
Applied mathematics: control theory in natural sciences; non-linear dynamics;
non-linear analysis of time series; fractional dimansional analysis;
computational topology in natural science models
Biomathematics: brain dynamics modeling; qEEG
Control in quantum physics.
Manipulation with quantum objects,including nanolithography and focusing, are natural and perspective areas of control application. We develop methods to control dynamics of quantum particles by the external optical and magnetic fields to focus them and to obtain the large angle coherent splitting effect for the initial wave packets.
Control algorithms allow to fabricate entangled qubit states. Our approach presents the linear control for different shapes of the external field to manipulate with the atomic level populations via the arbitrary shapes of the control signal. The feedback algorithm designs the classical external field for efficient control of qubit to achieve the desired state expressed with density matrix elements.
Cooling is important for the practical purposes of nanofabrication and for studyng the structure of atomic clusters and large molecules. We developed a model for sympathetic cooling of neutral particle (based on taking energy from atoms, molecules, or clusters by a relatively cold environment), expressing the time of cooling with the parameters of the quantum particle and the particles from the cooling reservoir.
We study the equation of energy spectrum balance for developed homogeneous isotropic turbulence.
The 1-D longitudinal energy spectrum calculated without fitting parameters is in good agreement with experimental data for the decaying case.
Biomathematics, Brain dynamics & Interdisciplinary research.
We perform non-linear analysis of time series and develop applications of mathematical methods of statistical physics and condensed matter to form a consistent hierarchic approach at micro-, meso- and macroscopic levels for brain dynamics to cover single neuran spike activity, neural clusters and dynamics of whole brain.
Our quantitative methods to analize EEG include statistical approaches, Lempel-Ziv complexity, fractional dimension analysis and computational topology.
We investigate the special classes of PDEs related to chaos theory and solitonic-type behavior.
One case is the multi-dimensional Schwarzian derivative of a real-valued function. Its basic properties related to its invariance under the group of multi-dimensional Möbius transformations. That allows to construct the solution to the n-dimensional Sturm-Liouville-like equations in Rn.
Another problem is the integrability of nonlinear partial differential equations with the special class of one-dimensional differential operators (‘Weiss operators’) that has been applied to solitonic-type PDEs. We investigate a solution of the Weiss operator family generalized for the case of Rn.