Mathematical Physics, Gravitation, and Solitons
Interaction between gravitational waves and matter
Dr. Davide Batic
- One of the most spectacular results of Einstein’s theory of gravitation is the theoretical prediction of gravitational waves which have been successfully detected by the LIGO and VIRGO experiments performed in 2015 and 2016, respectively. It is known that an electromagnetic wave generates a force acting on charged particles. Analogously, a gravitational wave exerts a force on massive particles. In this project, we study the effect of gravitational waves on the motion of Fermions and scalar particles by studying solutions of the Dirac and Klein-Gordon equations in a space-time described by the Minkwoski metric plus a small wave-like perturbation.
Light bending in a two-black hole metric
Dr. Davide Batic
- Light bending is maybe one of the most celebrated effects of General Relativity together with the recent discovery of the existence of Black Holes in our Universe. Inspired by this, we study a certain class of invariant hypersurfaces where closed photon circular orbits are allowed. More precisely, we focus on the weak and strong gravitational lensing in the presence of an accelerating black hole immersed in a universe with positive cosmological constant. Since the acceleration experienced by the black hole is due to the action of a cosmic string, these theoretical results might be compared with experimental values to probe into the existence of cosmic strings in our universe.
Gravitational lensing in the presence of dark matter halos.
Dr. Davide Batic
- It is well-known that light rays in the presence of a massive gravitational object such as for instance a star, a galaxy, or a black hole can be bended. More precisely, given certain density profiles associated with different dark matter models for galaxy halos, we derive the corresponding metrics for which we estimate the deflection angles for null rays in the strong and weak gravitational regimes. The results emerging from this study can be compared with existing astrophysical data, and this, in turn, allows to obtain further physical information on the properties of dark matter.
Estimation of Nonlinearities in optical Media
Dr. Berihu Teklu
- Quantum estimation is the theoretical tool to optimize quantum probes, i.e. quantum systems used as sensors because they are small, induce low disturbance and may be very precise since superpositions and entanglement are fragile features.
Mathematical modeling, control, and stability analysis of hybrid ODE-PDE systems under distributed loading
Dr. Alrazi Abdeljabbar and Dr. Muhammad Umer Shah
- This work addresses the problem of mathematical modeling, using Hamilton’s principle, and stability analysis, using the Lyapunov method, of a hybrid ODE-PDE system under distributed loading. The realization of the considered system is a flexible Cantilever beam transported using a lumped-mass system (i.e., an actuator) under distributed loads (i.e., wind or sea currents). This study aims at developing an effective feedback control strategy that can suppress the vibrations of the beam during its transportation and ensure a stable operation (i.e., the system should be at least uniformly ultimately bounded). Both rotational and translational motions of the beam are considered to address such problems in actual systems like manipulators and refueling machines. In this work, we also prove the well-posedness of the closed-loop system using the semigroup theory.
Exact Solutions to Nonlinear (2+1)-dimensional Partial Differential Equations
Dr. Alrazi Abdeljabbar
- In the past six decades, many powerful and systematic methods have been developed to obtain exact solutions for nonlinear differential equations which play an important role in soliton theory, such as inverse scattering method, the Darboux transformation, Bäcklund transformation, Hirota method and the Wronskian technique. The Wronskian technique has been applied to many soliton equations such as KdV, MKdV, NLS, derivative NLS, KP, sine-Gordon and sinsh-Gordon equations. Within Wronskian formulations, soliton solutions and rational solutions are usually expressed as some kind of logarithmic derivatives of Wronskian type determinants and the determinants involved are made of eigenfunctions satisfying linear system of differential equations. This connection between nonlinear problems and linear ones utilizes linear theories in solving soliton equations. In 1983, Nimmo and Freeman proved that the nonlinear Schrödinger equation has a double Wronskian solution. In view of some variants of Boussinesq systems studied before, we consider, in this project, (2+1)-dimensional system of nonlinear partial differential equations which can be considered as a generalization of (1+1)-dimensional variant Boussinesq model in the long gravity water waves.