I am an applied mathematician, and my research interests fall into the category of mathematical physics, with particular focus on
nonlinear partial differential and difference equations. Most of the models I work with are of the nonlinear wave type, meaning they
feature dispersion and nonlinearity. Nonlinear and linear wave equations can model a wide range of systems, including
optics, photonics, condensed matter, acoustics and water waves, and more. My work relies heavily on the synergy of analysis, computation and
experimentation. Links to all my papers (many with preprints) can be found on my publications page, while a short
synopsis of a few research themes is given
below. My work has been supported by the US National Science Foundation, Fulbright and internal Bowdoin grants.
A general audience summary of some of my prior and ongoing work can be found these articles:
Are you an undergraduate who is excited by fundamental science? Are you intrigued by the following you tube video of the inspirational Carl Sagan? Does working
on science homework not feel like work? If so, then you might be ready for an undergraduate research experience. I have many possible
student research projects, ranging in degree of difficulty. The minimum requirement is experience with ordinary differential equations (preferably as a full course, but
taken as part of an methods of physics course is also acceptable). Courses in numerical analysis, modeling, PDE and/or dynamical systems are a bonus.
I work with students in summer, or during the semester. If you are interested,
contact me. Here is a list of my past students and their projects:
Ari Geisler (Summer 2022) Title: Dispersive Shock Waves in Granular Crystals Funding: NSF grant DMS-2107945 Outcomes: Honors thesis and a peer reviewed article Summary:
Dispersive shock waves (DSWs), which connect states of different amplitude via an expanding wave train, are known to form in nonlinear dispersive media when subject to sharp changes in the initial state. Ari studied DSWs in a granular chain using numerical simulations and a long-wave length approximation.
Emmanuel Okyere (Summer 2022) Title: Solitary Waves in a Discrete Conservation Law Funding: NSF grant DMS-2107945 Outcomes: Peer reviewed article Summary: Discrete conservation laws can be used to model phenomena such as traffic flow. Emmanuel studied solitary waves in a discrete conservation law by first deriving a quasi-continuum approximation,
and finding exact solitary wave solutions of that model. He compared his analytical approximation against numerical solitary wave solutions of the discrete model.
Evy Wallace (Summer 2021) Title: Instability in a Time-Modulated Lattice Funding: NSF grant DMS-2107945 Outcomes: Honors thesis and a peer reviewed article Summary:Using Floquet theory and numerical simulation, Evy studied the stability of a nonlinear magnetic lattice with time-modulated stiffness. She was able to find a formula to predict the stability
in this lattice, and used that to generate stability diagrams in parameter space.
Bjorn Ludwig (Summer 2021 and 2022) Title: Nonlinear Resonance in a Membrane Funding: NSF grant DMS-2107945 Outcomes: Honors thesis
Summary:Bjorn studied nonlinear resonance in a membrane. In Summer 2021, he derived a planar ODE model for the description of the membrane and used fixed-point iterations and perturbation analysis to predict
the resonant peaks. In summer 2022, he used COMSOL to conduct finite element simulations of a nonlinear membrane, and to generate parameter values for his ODE prediction, and to compare
predictions between the high and low dimensional models.