Mark Hoefer

Office: ECOT 325

Dispersive Hydrodynamics Lab:  Muenzinger D047/D047a

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 including dispersive shock waves (DSWs) and solitary waves; 2) dynamics of ferromagnetic media in nanomagnetism. Methods employed include mathematical modeling, analysis, asymptotics, Whitham modulation theory, numerical analysis, and in-house experiment in the Dispersive Hydrodynamics Lab. Whenever possible, comparisons with experiment are carried out.

Dispersive Hydrodynamics Lab

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 currently supported by the National Science Foundation through DMS-1816934.

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.