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1. Mathematical methods in Physics: Integrable systems; Ordinary differential equations; Partial differential equations; Solutions of wave equations: bound states and solitons; integrable and near-integrable statistical physical models; condensed matter models and nonlinear partial differential equations in general physics: the exact solutions, solitons, instantons, and asymptotic solutions.

2. Matter waves, quantum condensation phenomena and super-fluidity: Matter waves Bose-Einstein condensates; Bose-Einstein condensates in periodic potentials; solitons; vortices; Gross-Pitaevskii equations.

The wavefunction of the Bose-Einstein
condensate is mainly governed by the Gross-Pitaevskii (GP) equation. Ruprecht
et al. have solved the time-dependent, driven GP equation for the BEC by
direct numerical integration. With increasing strength of the perturbation, the
nonlinear response of the condensate gives rise to the generation of harmonics
of the driving frequency and frequency mixing between the normal modes. The
origin of this nonlinear response is of course the nonlinear mean-field
potential; a perturbation of the overall potential causes a time-dependent
change in the wavefunction, which in turn causes a change in the nonlinear
potential, and so on. The nonlinear phenomena that result are matter-wave
analogs of the corresponding effects arising in conventional nonlinear optics. Based
on the GP equation, we aim in this research to study nonlinear phenomena in
BEC. These nonlinear phenomena include bright, dark, gap and multidimensional
solitons; vortices; vortex lattices; optical lattices; multicomponent
condensates; manipulation of condensates. Both theoretical and experimental
studies will be carried out.

Nonlinear
transmission lines are used for pulse shaping. In my works on this area of
research, we generally develop the theory of pulse propagation through the NLTLs.
Using a gradually scaled NLTL, we solve the problem of a wide pulse
degenerating into multiple pulses rather than a single pulse. The possibility
of soliton propagation on the NLTLs is widely studied in our works. In fact, b*y using **nonlinear transmission
lines** *as a
convenient tool to study wave propagations in dispersive media, we have solved various
soliton *problems.** *

Working in the
field of differential equations, the boundary value problems in our interest is a differential equation together with a set of
additional restraints, called the boundary
conditions. This class of problems arises in several branches of physics
as any physical differential equation will have them! In practice, any boundary
value problem can be useful if and only if it is well-posed, i.e., if its solution
exists, it is unique and depends continuously on data of the problem. Using the
theory of generalized functions and the theory of distribution, we investigate
in our work the conditions of the well-posedness of boundary value problems for
partial differential equations of mathematical physics. With VM. Borok, I have
introduced in 1994 (for the first time) the concept of asymptotical
well-posedness in mathematical literature.

5. Biomathemacs (Problems of bio-heat transfer in biological systems): Problems of bio-heat transfer in biological systems have significant applications in a wide variety of clinical, basic, and environmental sciences. In particular, understanding the heat transfer in biological tissues is a necessity for many therapeutic practices involving either raising or lowering temperature such as cancer hyperthermia, burn injury, brain hypothermia resuscitation, disease diagnostics, thermal comfort analysis, cryosurgery and cryopreservation, and so on. In my works on this area, an overview on the most basic bio-heat transfer model, its extended forms as well as typical applications under various high- or low-temperature situations is summarized. The model I generally use is more complex than the one proposed by Pennes. Its complexity is due to the fact that the blood perfusion is not more a constant, but depends implicitly on the temperature. Therefore, the model describing temperature distribution in the living tissue is governed by a strongly nonlinear partial differential equation of Pennes type. Because on the nonlinearity, solutions to our bio-heat transfer model are generally numerical. Some possible extensions of the classic bioheat transfer model are also discussed.