Applied Mathematics, Mathematical Physics, Nonlinear Waves, Parity-Time Symmetry, Scientific Computing, Sum-of-squares Optimization
Buffalo NY, 14260-2900
Phone: (716) 645-8814
Postdoc University of Victoria, Canada
Postdoc University of Cape Town, South Africa
Ph.D Mathematics, McMaster University, Canada
B.Sc. Information and Computer Science, Moscow Institute of Electronic Technology, Russia
Tuesdays 2pm-3pm, Thursdays 10am-11am
My research focus lies in Applied Mathematics broadly; more precisely, I study existence and stability of nonlinear waves in ordinary and partial nonlinear differential equations inspired by real-world applications (ODEs and PDEs, correspondingly), both numerically and analytically. The main applications of the type of equations I study lie in quantum mechanics (finding new memory types for quantum computers), nonlinear optics (building novel laser devices and fibers), material science (developing meta-materials with desired properties), as well as other areas of Physics. The following are the main themes in my work.
Solitary wave solutions (solitons) were first observed in the water by John Scott Russell in 1834 and nowadays attract a lot of attention in many nonlinear physical and biological systems. Solitons require a balance between dispersion and nonlinearity and usually form families with different amplitudes. The importance of solitons is underlined by the so-called 'soliton resolution conjecture': roughly speaking, it says that all reasonable (finite energy) solutions to nonlinear dispersive PDEs resolve into a superposition of radiation (which behaves like a solution of a linear Schroedinger equation) and a finite number of solitons.
PT-symmetric systems are open systems where dissipative losses are exactly compensated by symmetrically arranged energy gain: they preserve space reflection P and time-reversal along with complex conjugation T symmetries simultaneously. Such systems appeared in quantum mechanics as an attempt to relax Hermiticity property of operators describing particles. A PT-symmetric operator can have real spectrum in a certain regime, despite not being Hermitian, and can describe unitary time evolution, both essential properties in quantum mechanics. These days, most applications of the concept appear in nonlinear optics, where PT-symmetric systems were experimentally observed. The majority of discovered PT-symmetric equations are variations of the nonlinear Schroedinger equation.
Convex optimization and sum-of-squares programming, coming the field of real algebraic geometry, have recently found many applications in analysis of ODEs and PDEs. When the underlying equations are formulated in terms of polynomials, one can formulate sum-of-squares auxiliary functions using computer-assisted methods, and study existence of traveling waves, nonlinear stability questions through constructing Lyapunov functions, as well as finding basins of attraction, and much more. Sum-of-squares problems can be solved numerically via semidefinite programming, a step up from linear programming where approximate solutions can be found using modern interior-point methods.