About

Mark Hoefer is a Professor and Department Chair in Applied Mathematics at the University of Colorado Boulder. His research is centered on physical applied mathematics motivated by real-world problems, with a focus on nonlinear waves. His work primarily involves the fluid dynamics of dispersive media, including dispersive shock waves (DSWs) and solitary waves, as well as the dynamics of ferromagnetic media in nanomagnetism. Hoefer employs a variety of methods such as mathematical modeling, analysis, asymptotics, Whitham modulation theory, numerical analysis, and in-house experiments within the Dispersive Hydrodynamics Lab. His research on DSWs explores their role as a universal mechanism to resolve hydrodynamic singularities in dispersive media, with physical manifestations including undular bores, nonlinear diffraction patterns, and matter waves. Additionally, Hoefer investigates nonlinear, dispersive phenomena in ferromagnetic media, particularly the excitation of magnetization dynamics at the nanometer scale using spin polarized currents, leading to the observation of strongly nonlinear magnetic solitons or 'droplets' in nanomagnetic systems.

Research topics

• Physics
• Mechanics
• Quantum mechanics
• Optics
• Mathematics
• Classical mechanics
• Mathematical analysis
• Mathematical physics

Selected publications

• Homoclinic and heteroclinic solutions of the nonlinear Schrödinger equation with a complex Wadati potential

  arXiv (Cornell University) · 2026-04-09

  preprintOpen accessSenior author

  Stationary solutions asymptoting to nonlinear plane waves of the nonlinear Schrödinger equation with a PT-symmetric, complex linear potential are characterized. The potential includes both a spatially varying gain-loss profile and a repulsive real part, generated by a Wadati potential function,that support the existence of homoclinic and heteroclinic solutions that asymptote to the same or different, respectively, nonlinear plane waves in the far field. Asymptotic analysis and numerical simulations are used to examine solution existence, bifurcations, and structure. Such solutions play an important role in resonant nonlinear wave generation of dispersive media with localized gain and loss.

  PublisherDOI

• Whitham modulation equations for the regularized Boussinesq equation with cubic nonlinearity

  ArXiv.org · 2026-05-13

  articleOpen access1st authorCorresponding

  A regularized Boussinesq equation is studied as a dispersive, long-wave (quasicontinuum) approximation of the Fermi-Pasta-Ulam lattice with a general cubic interaction force. Explicit periodic traveling wave solutions in terms of Jacobi elliptic functions are classified, and their solitary-wave, kink, and trigonometric limits are obtained. The Whitham modulation equations describing slow modulations of periodic traveling wave solutions are derived using an averaged variational principle. The convexity (strict hyperbolicity, genuine nonlinearity) of the resulting hydrodynamic-type equations is examined numerically in general and analytically in the solitary-wave and harmonic limits. In particular, the loss of hyperbolicity and the formation of complex conjugate characteristic speeds is shown to lead to modulational instability of periodic traveling waves. The onset of modulational instability is verified by numerical computations of linearized spectra for periodic traveling waves and initial value problems that also reveal additional short-wave instabilities.

  PublisherOA PDF

• Homoclinic and heteroclinic solutions of the nonlinear Schrödinger equation with a complex Wadati potential

  arXiv (Cornell University) · 2026-04-09

  articleOpen accessSenior author

  Stationary solutions asymptoting to nonlinear plane waves of the nonlinear Schrödinger equation with a PT-symmetric, complex linear potential are characterized. The potential includes both a spatially varying gain-loss profile and a repulsive real part, generated by a Wadati potential function,that support the existence of homoclinic and heteroclinic solutions that asymptote to the same or different, respectively, nonlinear plane waves in the far field. Asymptotic analysis and numerical simulations are used to examine solution existence, bifurcations, and structure. Such solutions play an important role in resonant nonlinear wave generation of dispersive media with localized gain and loss.

  PublisherOA PDF

• Whitham modulation equations for the regularized Boussinesq equation with cubic nonlinearity

  arXiv (Cornell University) · 2026-05-13

  preprintOpen access1st authorCorresponding

  A regularized Boussinesq equation is studied as a dispersive, long-wave (quasicontinuum) approximation of the Fermi-Pasta-Ulam lattice with a general cubic interaction force. Explicit periodic traveling wave solutions in terms of Jacobi elliptic functions are classified, and their solitary-wave, kink, and trigonometric limits are obtained. The Whitham modulation equations describing slow modulations of periodic traveling wave solutions are derived using an averaged variational principle. The convexity (strict hyperbolicity, genuine nonlinearity) of the resulting hydrodynamic-type equations is examined numerically in general and analytically in the solitary-wave and harmonic limits. In particular, the loss of hyperbolicity and the formation of complex conjugate characteristic speeds is shown to lead to modulational instability of periodic traveling waves. The onset of modulational instability is verified by numerical computations of linearized spectra for periodic traveling waves and initial value problems that also reveal additional short-wave instabilities.

  PublisherDOI

• Solitary wave-mean flow interaction in strongly nonlinear dispersive shallow water waves

  Journal of Nonlinear Waves · 2025-01-01 · 2 citations

  preprintOpen access

  Abstract The interaction of a solitary wave and a slowly varying mean background or flow for the Serre-Green-Naghdi (SGN) equations is studied using Whitham modulation theory. The exact form of the three SGN-Whitham modulation equations – two for the mean horizontal velocity and depth decoupled from one for the solitary wave amplitude field – is obtained in the solitary wave limit. Although the three equations are not diagonalizable, the restriction of the full system to simple waves for the mean equations is diagonalized in terms of Riemann invariants. The Riemann invariants are used to analytically describe the head-on and overtaking interactions of a solitary wave with a rarefaction wave and dispersive shock wave (DSW), leading to scenarios of solitary wave trapping or transmission by the mean flow. The analytical results for overtaking interactions prove that a simpler, approximate approach based on the DSW fitting method is accurate to the second order in solitary wave amplitude, beyond the first order accurate Korteweg-de Vries approximation. The analytical results also accurately predict the SGN DSW’s solitary wave edge amplitude and speed. The analytical results are favourably compared with corresponding numerical solutions of the full SGN equations. Because the SGN equations model the bi-directional propagation of strongly nonlinear, long gravity waves over a flat bottom, the analysis presented here describes large amplitudesolitary wave-mean flow interactions in shallow water waves.

  PublisherDOI

• Mach Reflection and Expansion of Two-Dimensional Dispersive Shock Waves

  Physical Review Letters · 2025-07-02 · 4 citations

  articleSenior author

  The oblique collisions and dynamical interference patterns of two-dimensional dispersive shock waves are studied numerically and analytically via the temporal dynamics induced by wedge-shaped initial conditions for the Kadomtsev-Petviashvili II equation. Various asymptotic wave patterns are identified, classified, and characterized in terms of the incidence angle and the amplitude of the initial step, which can give rise to either subcritical or supercritical configurations, including the generalization to dispersive shock waves of the Mach reflection and expansion of viscous shocks and line solitons. An eightfold amplification of the amplitude of an obliquely incident flow upon a wall at the critical angle is demonstrated. Applications of the results include bore interactions in geophysical fluid dynamics.

  PublisherDOI

• Breaking a Superfluid Harmonic Dam: Observation and Theory of Riemann Invariants and Accelerating Sonic Horizons

  Physical Review Letters · 2025-11-07 · 1 citations

  articleOpen access

  An experimental and theoretical study of sonic horizons emerging from the dam-break problem in a Bose-Einstein condensate confined in an anisotropic harmonic trap is presented. Measurements, analysis, and numerics reveal the formation of a sonic horizon that undergoes acceleration due to harmonic confinement. The superfluid is characterized using a robust measurement technique to determine Riemann invariants. Experimental observations agree with an analytical solution of the Gross-Pitaevskii equation and computations. The collision and annihilation between two sonic horizons at long times is predicted.

  PublisherDOI

• Bright traveling breathers in media with long-range nonconvex dispersion

  Physical review. E · 2024-03-27 · 1 citations

  articleSenior author

  The existence and properties of envelope solitary waves on a periodic traveling-wave background, called traveling breathers, are investigated numerically in representative nonlocal dispersive media. Using a fixed-point computational scheme, a space-time boundary-value problem for bright traveling breather solutions is solved for the weakly nonlinear Benjamin-Bona-Mahony equation, a nonlocal, regularized shallow water wave model, and the strongly nonlinear conduit equation, a nonlocal model of viscous core-annular flows. Curves of unit-mean traveling breather solutions within a three-dimensional parameter space are obtained. Resonance due to nonconvex, rational linear dispersion leads to a nonzero oscillatory background upon which traveling breathers propagate. These solutions exhibit a topological phase jump and so act as defects within the periodic background. For small amplitudes, traveling breathers are well approximated by bright soliton solutions of the nonlinear Schrödinger equation with a negligibly small periodic background. These solutions are numerically continued into the large-amplitude regime as elevation defects on cnoidal or cnoidal-like periodic traveling-wave backgrounds. This study of bright traveling breathers provides insight into systems with nonconvex, nonlocal dispersion that occur in a variety of media such as internal oceanic waves subject to rotation and short, intense optical pulses.

  PublisherDOI

• Mach reflection and expansion of two-dimensional dispersive shock waves

  arXiv (Cornell University) · 2024-11-08

  preprintOpen accessSenior author

  The oblique collisions and dynamical interference patterns of two-dimensional dispersive shock waves are studied numerically and analytically via the temporal dynamics induced by wedge-shaped initial conditions for the Kadomtsev-Petviashvili II equation. Various asymptotic wave patterns are identified, classified and characterized in terms of the incidence angle and the amplitude of the initial step, which can give rise to either subcritical or supercritical configurations, including the generalization to dispersive shock waves of the Mach reflection and expansion of viscous shocks and line solitons. An eightfold amplification of the amplitude of an obliquely incident flow upon a wall at the critical angle is demonstrated.

  PublisherOA PDFDOI

• Two-dimensional reductions of the Whitham modulation system for the Kadomtsev–Petviashvili equation

  Nonlinearity · 2024-01-18 · 1 citations

  articleOpen access

  Abstract Two-dimensional reductions of the Kadomtsev–Petviashvili(KP)–Whitham system, namely the overdetermined Whitham modulation system for five dependent variables that describe the periodic solutions of the KP equation, are studied and characterized. Three different reductions are considered corresponding to modulations that are independent of x , independent of y , and of t (i.e. stationary), respectively. Each of these reductions still describes dynamic, two-dimensional spatial configurations since the modulated cnoidal wave, generically, has a nonzero speed and a nonzero slope in the xy plane. In all three of these reductions, the integrability of the resulting systems of equations is proven, and various other properties are elucidated. Compatibility with conservation of waves yields a reduction in the number of dependent variables to two, three and four, respectively. As a byproduct of the stationary case, the Whitham modulation system for the classical Boussinesq equation is explicitly obtained.

  PublisherOA PDFDOI

Recent grants

• Conference: Emergent Phenomena in Nonlinear Dispersive Waves

  NSF · $15k · 2024–2025

• Dispersive Hydrodynamics and Applications

  NSF · $434k · 2018–2023

• Supersonic Dispersive Fluid Dynamics

  NSF · $131k · 2010–2014

• CAREER: Solitary Waves and Wavetrains in Dispersive Media

  NSF · $187k · 2013–2015

• Nonlinear Wave Interactions

  NSF · $300k · 2023–2027

Frequent coauthors

• Ezio Iacocca

  University of Colorado Colorado Springs

  31 shared

• G. A. Él

  Northumbria University

  22 shared

• Michelle Maiden

  University of Colorado Boulder

  18 shared

• Peter Engels

  Washington State University

  18 shared

• Patrick Sprenger

  University of California, Merced

  18 shared

• T. J. Silva

  NAVSYS (United States)

  14 shared

• Gino Biondini

  10 shared

• Nicholas K. Lowman

  10 shared

Education

• PhD in Applied Mathematics, Applied Mathematics

  University of Colorado Boulder

  2006

• MS in Applied Mathematics, Division of Engineering and Applied Sciences

  Harvard University

  2000

• BS in Mathematics of Computation, Mathematics

  University of California Los Angeles

  1997