In general, my research lies at the interface of mathematics and real-world physical phenomena. It consists of modeling physical systems in terms of ordinary and partial differential equations and employing functional analysis, asymptotic and perturbation analysis, and numerical computations to analyze the models in detail. The over-arching goal of my research is to connect between the mathematical and physical aspects arising from these problems and make reliable and useful predictions about physical systems.
Since their invention in the 1960s, lasers have had a profound impact on almost every aspect of modern science and technology, from fundamental physics to telecommunications to medicine. My research results have application to such areas as design and operation of ultrafast mode-locked lasers and the propagation of laser beams in optical fibers, air and water. My research aims to help answer fundamental questions, such as how do these systems behave and how can their operation be improved? I am also interested in the dynamics of ultra-cold atomic gases. When matter is cooled down very near the absolute zero temperature, a large fraction of the atoms collapse into the lowest allowable quantum state, giving rise to quantum effects on a macroscopic scale (Bose-Einstein condensates).
In-spite of their apparent differences these physical systems have much in common. One of their remarkable features is that nonlinear waves (often called solitons or solitary waves) can propagate through them almost unscathed. On the other hand, these waves can also undergo very intense phenomena, such as self-focusing (wave collapse). Mathematically, such systems can ostensibly be modeled by universal types of mathematical equations, in particular by nonlinear Schrödinger (NLS)-type equations.