Mark Hoefer

Research

My research is in physical applied mathematics, centered on nonlinear waves. Most of my work is in the field of dispersive hydrodynamics: how a large collection of waves move through a medium when nonlinearity and wave dispersion together govern its motion. Water waves, superfluids, magnetic materials, optical media and discrete lattices all exhibit this behavior, and I look for the mathematical structures they share. My group combines asymptotic analysis and Whitham modulation theory with numerical simulation and laboratory experiments in the Dispersive Hydrodynamics Laboratory. We test theory directly against observation.

Dispersive shock waves and modulation theory. In a dispersive medium, a would-be shock (steep gradient) resolves into an expanding, coherent train of oscillations called a dispersive shock wave (DSW). Examples include undular bores in rivers and the atmosphere, quantum shock waves, and nonlinear diffraction in optics. Whitham modulation theory is our main analytical tool for describing these structures, and we are extending it in several directions:

• Higher dimensions. We developed modulation theory for the Kadomtsev–Petviashvili (KP) and multidimensional nonlinear Schrödinger equations. This led to a theory of Mach reflection and expansion of two-dimensional DSWs, of soliton resonance, and of KP lumps interacting with a mean field. We are also studying Mach reflection in the two-dimensional Benjamin–Ono equation.
• Nonconvex systems. We study equations with nonconvex, higher order or full dispersion and equations with nonconvex hydrodynamic flux.  These systems result in new, nonclassical types of DSWs.
• Discrete systems. We study the hydrodynamics of lattice conservation laws, DSWs in Fermi–Pasta–Ulam–Tsingou chains, and quasi-continuum models.

Solitons, mean flows and integrable turbulence. A central theme of the group is how solitary waves interact with large-scale "mean flows." Solitons can be trapped by, tunnel through, or be refracted by an evolving large-scale background such as a DSW or a rarefaction wave.  We have developed this theory for a variety of dispersive hydrodynamic equations.  We have used the kinetic theory of soliton gases to model statistical ensembles of interacting waves. This has produced an exactly solvable model of wave–mean field interaction in integrable turbulence.

Breathers and coherent structures. We study traveling breathers, which are localized time-periodic waves on an oscillatory background. These include KdV breathers riding on periodic cnoidal backgrounds. In a two-fluid system (viscous fluid conduits), we demonstrated the first laboratory observation of traveling breathers and their scattering.

Superfluid and quantum hydrodynamics. We work with experimental collaborators in ultracold atomic physics, using Bose–Einstein condensates as an ideal testbed for dispersive hydrodynamics. Recent work combines theory and observation of a superfluid "dam break" in a harmonic trap. It shows Riemann invariants directly and reveals accelerating sonic horizons. Related work studies dissipative shocks driven by a quantum-mechanical piston. Ongoing work uses three-dimensional Gross–Pitaevskii simulations to study quantum turbulence in elongated condensates.

Magnetic materials and spin hydrodynamics. In ferromagnetic thin films, we model magnetization dynamics as a spin superfluid. This framework predicts spin-injection-driven shock waves and solitons, and long-distance spin transport through noncollinear states. Related work covers:

• magnetoelastic waves in layered materials
• ultrafast optical perturbation of magnetic domains
• energetics of phase transitions in antiferromagnets
• our earlier work on magnetic droplet solitons

Laboratory experiments. In viscous core-annular flows, a buoyant fluid rises through a heavier, more viscous one, and the interface carries clean, controllable nonlinear waves. This makes it a laboratory playground for dispersive hydrodynamics. These conduits let us create DSWs, solitary-wave fission, periodic traveling waves, wavemaker-driven radiation and breathers on demand, and compare them quantitatively against theory.

This work is supported by the National Science Foundation under grant DMS-2306319, Nonlinear Wave Interactions.