RESEARCH  PROJECTS  FOR  MSc/PhD  STUDENTS

Prerequisites

For working with me, students must have a good theoretical background. Knowledge of some nonlinear physics is an advantage. Experience with programming is also an advantage.

Research Directions

    Below is a brief description of two of my current research directions. Both belong to the developing field of theoretical "Nonlinear Science" and deal with understanding of formation and control of coherent nonlinear structures (patterns) in extended physical systems. Different topics in each direction can form a basis for MSc or PhD theses depending on the scope of research.

I. Autoresonance of Coupled Nonlinear Waves (MSc/PhD)

Introduction. Controlled excitation of nonlinear waves in fluids and plasmas is one of the most important goals in both basic and applied research. Autoresonance (adiabatic nonlinear synchronization) is a convenient general approach to achieving this goal, as demonstrated in several recent applications. The method involves an extended system in some initial equilibrium, driven by an external wave-like perturbation having slowly varying frequency and/or wave vector. The driven nonlinear wave emerges when the drive, assumed to be independent of the driven wave in most applications, passes a resonance with the system. At certain conditions, the excited wave phase-locks with the drive in an extended region of space-time (the autoresonance phenomenon) and its amplitude may grow significantly as the driving wave parameters vary in space and/or space.

Proposed Research. It is proposed to add self consistently in this problem and study a number of important applications involving autoresonance of mutually interacting nonlinear waves. In particular, we shall investigate autoresonant Ramann and Brillouin scattering in space-time varying plasmas, control of large amplitude diocotron waves and other rotating equilibria in pure electron plasmas by three wave mixing, and, more generally, study adiabatic synchronization of coupled single and multi-phase nonlinear waves described by some of the most fundamental models of nonlinear physics, i.e., the Korteweg-de-Vries (KdV) and Nonlinear Schrödinger (NLS) equations. The proposed research will comprise an important contribution to our understanding of synchronization of nonlinear waves in space-time varying media with the potential for many new applications in plasma physics and related fields, such as nonlinear optics, hydrodynamics, and acoustics.

II. Resonant Kinetics and Driven Plasma and Vorticity Excitations (MSc/PhD)

Introduction. Frequently, waves in plasmas are described by the fluid approximation. Such fluid modes have been studied extensively both experimentally and theoretically. On the other hand, nearly 50 years ago, Bernstein, Greene and Kruskal (BGK) predicted existence of a large class of purely kinetic, dissipationless waves in plasmas, but experimental realization of these BGK modes is known to be difficult. A significant progress in the field was reported in recent experiments and theory on excitation of large amplitude chirped-bucket BGK (CBGK) modes in pure electron plasmas. The CBGK modes were excited by capturing a group of plasma electrons in the tail of the velocity distribution into resonance with external potential oscillations and dragging the trapped particles into the bulk of the velocity distribution by slowly varying the frequency of the external perturbation. The relocation of a low density region in phase-space created a growing depression (hole) in the phase-space distribution of plasma particles as this region entered the bulk of the distribution. The CBGK modes are intrinsically nonlinear plasma density/field structures associated with these phase-space holes.
Proposed Research. We propose to significantly broaden our theoretical understanding of CBGK excitations and apply similar ideas in other directions in plasmas and fluids. Particularly, we shall focus on wave-plasma interactions known as Ramann and Brillouin scattering. Conventionally, one of the channels in these 3-wave interactions is a fluid-type wave in plasma. We shall study a different scheme of Ramann and Brillouin scattering with the fluid waves replaced by CBGK modes. The latter will emerge as the result of the 3-wave interaction process. Furthermore, we shall use the similarity between the 1D plasma system describing the CBGK modes and equations for 2D ideal fluids in investigating new m-fold symmetric CBGK-type vortices (V-states). Analogous structures (patterns) will be discussed in the context of the ExB drift dynamics in magnetized nonneutral plasmas (numerous experiments with these plasmas are performed by my collaborator, Prof. J. Fajans at UC Berkeley). We shall study excitation dynamics, stability and control of these V-states. We expect that the proposed study of CBGK-type excitations will lead to a better understanding of general wave interactions involving resonant kinetics in plasmas and fluids with a potential of many new applications.