Since 2010, when I became an invited professor at Linköping University, I have been working within the following topics:
1. Modeling of circulatory blood system, in particular arterial and venous trees.
- two-dimensional model of elastic walls of blood vessels;
- one-dimensional model of interaction between blood flow (viscous and visco-elastic fluid) with elastic wall of blood vessel;
- the transmission conditions in the one-dimensional model of bifurcation nodes of arteries, namely substantial modifications of the classical Kirchhoff transmission conditions which takes into account elastic properties of walls and spacial shape of nodes.
2. Wave processes in cylindrical and periodic waveguides, acoustic, quantum, elastic and piezoelectric as well as water-waves.
- a criterion for the existence of trapped modes and the notion of enforced stability of embedded eigenvalues, in particular for graphen ribbons;
- asymptotic description of spectral bands and spectral gaps in periodic waveguides and lattices;
- the notion of "almost standing" waves at threshold frequencies and explanation of near-threshold anomalies generated by these waves;
- development of do-all mathematical tool to impose the energy radiation conditions based on the Umov - Poynting - Mandelstam principle;
- the notion of "wandering" eigenvalues and description of a novel approach to form the continuous spectrum.
3. Spectral problems in domains with cuspidal singularities on the boundary.
- a mathematical explanation of the effect of "black holes" for acoustic and elastic waves;
- a criterion for the nonempty continuous spectrum in general elliptic problems in domains with cuspidal singularities;
- asymptotic description of "wandering eigenvalues" due to smoothing of the cuspidal singularity.