## Research Interests

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.
A general audience summary of some of my prior and ongoing work is featured in the following
article of the Bowdoin Daily Sun.

My research is supported by the NSF through grant DMS-1615037.

## Undergraduate Research

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:

**Madeleine Genereux ’19 and Sean McParland '19**

Title: Approximation of Discrete DSWs through PDE approximations

(supported by the NSF through grant DMS-1615037)

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. Such extreme changes in a medium can occur when simulating explosions in shock tubes, when gasses or fluids are compressed in a piston chamber, or when water dams break. Sean and Madeline investigated DSWs in a system consisting of masses coupled via nonlinear springs. Using a multiple-scale analysis, they were able to derive PDEs, and from there to approximate the formation and structure of the DSWs. They also conducted numerical simulations to investigate the dynamics.

**Kevin Chen '19 **

Title:Parametric Instability in Granular Chains with Time-Dependent Stiffness
(supported by the NSF through grant DMS-1615037)

A granular chain is a one dimensional network of particles that interact elastically. Kevin studied a granular chain where the stiffness of the particles would vary in time. In particular, the stiffness would vary periodically (according to a step-function). He explored how instabilities can be introduced into the system via variable stiffness through the computation of the dispersion relationship and numerical simulations.

**Waverly Harden ’19**

Title: Dynamics of Hollow Elliptical Cylinder Arrays

(stipend and travel support by NSF through grant DMS-1615037, additional travel support provided by the Grua/O’Conell Fund)

Strain-softening materials have gained significant interest in the physics, mathematics and material science communities recently. Most materials that we are familiar with
are strain-hardening, meaning as they deform, they become harder to deform. Strain-softening materials have
the opposite property; as they deform, they become easier to deform. The dynamics of strain-softening materials in an experimental setting have been rarely studied.
For seven weeks, Waverly worked on the modeling and simulation of a system that exhibits strain-softening behavior. It consists of a chain of hollow elliptical cylinders (HEC).
She visited the laboratory of her advisors collaborator, Professor Jinkyu Yang, at the University of Washington for the final three weeks of her project to work with the team who developed
the concept of the HEC and the experimental set-up. Using the experimental data, she was able to refine her mathematical model. She also worked on an experimental apparatus
that controls the impact frequency of the chain, and worked on processing of the experimental data that was acquired through digital image correlation.

**Jacob Hart ’17 and Carina Spiro ’18 **

Title: Nonlinear Energy Harvesting

(stipends supported by the Maine Space Grant Consortium, travel supported by NSF through grant DMS-1615037)

The last several decades as seen a burst of research activity in vibration energy harvesting, which deals with the conversion of mechanical energy (such as
vibration) into electrical energy. Most commercial energy harvesters are based on linear responses, which have a relatively restricted range of vibration frequencies
where energy harvesting is efficient. Jake and Carina explored the integration of novel aspects of nonlinear wave equations to try to harvest energy in new ways. In particular, they studied an array of cantilevered beams
that are coupled via magnetic links. For the first six weeks of the project, they worked on the modeling of the system, its numerical simulation, and the computation of periodic
orbits. They visited the laboratory of their advisors collaborator, Professor Chiara Daraio, at the ETH Zurich for their final two weeks to work with the team developing the
experimental set-up of the array of cantilevers. Using the experimental data, they were able to refine their mathematical model.

## Summary of Some Work

#### Breathers in Granular Crystals

Some of my current work involves localized structures in granular chains. An example of a granular crystal is shown on the left above (© Boechler/Theocharis Caltech 2011). A numerical solution of the relevant equations of motion is shown on the right. Such a solution is called a discrete dark breather, which is time periodic solution with a nonzero background. We work closely with several experimental groups, including

- Nicholas Boechler, Mechanical Engineering, UC San Diego
- Chiara Daraio, Mechanics and Materials, Caltech
- Georgios Theocharis, CNRS, Le Mans France
- Jinkyu Yang, Aeronautics & Astronautics, Univ. Washington

#### Justification of Amplitude Equations

One focus of mine deals with the justification of amplitude equations. In the image above, solutions to a periodic Fermi-Pasta-Ulam lattice are shown (makers), with amplitudes approximately described by the Korteweg-de Vries equation (left) and the the nonlinear Schrödinger equation (right). Justification in this sense means that the error of such approximations is sufficiently small, see [10] for more details.

Another example is the justification of the NLS equation as an amplitude equation for semilinear wave equations with unstable quadratic resonances and for quasilinear systems with non-resonant quadratic terms. This research was conducted under the grant (Schn 520/8-1) which is sponsored by the German Research Foundation (DFG) and was headed by Guido Schneider.

#### Pulse Interaction

Shown above is a movie of two pulse evolution in the Klein-Gordon equation with a cubic nonlinearity
with rectangular (top) and NLS 1-soilton (bottom) envelopes.
See the paper [3] for more details.
The code to generate such solutions can be found in the
code section below.

#### Discrete Solitons

The discrete NLS equation (DNLS) with extended linear coupling is known to support discrete solitons (pictured left). Unlike in the standard DNLS equation, these solitons have complex valued amplitudes. Such solutions can be approximated using a variational approximation (VA), which reduces the infinite set of equations to a small (albeit complicated) system. Within this approximation, complex bifurcation scenarios can be captured. In the right panel above, the relative phase difference of adjacent nodes of the soliton solutions in the left panel are shown for exact solutions (colored lines) and the VA (black lines). See [7] for details. More recently, my collaborators and I have shown that the variational approximation can be justified rigorously [9]

The image above shows the snaking structure of the discrete cubic-quintic NLS equation. The left panel shows the norm of various discrete solitons. An example of one solution is given in bottom right panel with the corresponding linear stability spectrum in the top right panel. See papers [1,4,6] for more details.