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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.3. Nonlinear waves: Nonlinear transmission lines; Solutions of nonlinear wave equations; Nonlinear Schrödinger equations; Ginzburg-Landau equations; Numerical solutions of nonlinear partial differential equations; Solitary wave solutions of nonlinear partial differential equations.
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, by using nonlinear transmission lines as a convenient tool to study wave propagations in dispersive media, we have solved various soliton problems.4. PDE of the mathematical physics: Boundary value problems for PDE; Well posedness of boundary value problems; Problems of small denominators; optimal control.
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.