Research in physical applied mathematics, motivated by real world
problems is central to my investigations. My research is generally in
the field of nonlinear waves with a focus on two areas: 1) fluid
dynamics of dispersive media with accompanying wave excitations
shock waves (DSWs) and solitary waves; 2) dynamics of ferromagnetic
media, spin torque, and localized excitations in nanomagnetism.
Methods employed include mathematical modeling, analysis, asymptotics,
Whitham modulation theory, and numerical analysis. Whenever possible,
comparisons with experiment are carried out.
The generation of DSWs represents a universal mechanism to resolve
hydrodynamic singularities in dispersive media. Physical
manifestations of DSWs include undular bores on shallow water and in
the atmosphere (the Morning Glory), nonlinear diffraction patterns in
optics, and matter waves in ultracold atoms. Any approximately
conservative, nonlinear, hydrodynamic medium exhibiting weak
dispersion can develop DSWs. The mathematical description of DSWs
involves a synthesis of methods from hyperbolic quasi-linear systems,
asymptotics, and soliton theory. This research
is supported by the National Science Foundation through grants
DMS-1008973 and CAREER DMS-1255422.
Ferromagnetic media provide a source of rich nonlinear, dispersive
phenomena with practical import. Theoretical and technological
developments have stimulated the field of nanomagnetism by the
introduction of spin polarized currents as a means to excite
magnetization dynamics at the nanometer scale in patterned
environments. Strongly nonlinear magnetic solitons were
observed in a nanomagnetic system. This solitary wave or
"droplet" joins the domain wall and magnetic vortex as a fundamental
and distinct object in nanomagnetism with similar potential for
fruitful science. This research is supported by the National Science
Foundation through a Career award, DMS-1255422.
Seeking undergraduate and current/prospective
graduate students interested in studying nonlinear waves and
applications. Funding may be available.
Please contact me if you are