I am a professor in the Department of Mathematical Sciences at NJIT. My research applies methods of dynamical systems to (conservative, dispersive) nonlinear wave equations arising, for example, in optics and fluid mechanics. I am interested in dynamical and bifurcation phenomena, and I use techniques from Hamiltonian mechanics and perturbation theory, as well as extensive computations. My current research in vortex dynamics is supported by the NSF under DMS-2206016.
I led the effort to start the SIAM New York-New Jersey-Pennsylvania Section (SIAM NNP) and was its first president in 2023-2024. I have handed over the reins to a new leadership group.
Interests
- Nonlinear Waves
- Dynamical Systems
- Asymptotics and Perturbation Methods
- Scientific Computing
- Point Vortices
Education
- PhD, Mathematical Sciences, 1999New York University
- BS, Mathematics, 1994University of Michigan
Recent Publications
Phase portraits and the bifurcation set for the three-vortex interaction system
Atul Anurag (अतुल अनुराग) and Roy Goodman. Communications in Nonlinear Science and Numerical Simulation, 2026.
We use Jacobi coordinates and a momentum map/Lie-Poisson reduction to derive a system of equations for the three-vortex problem with no coordinate singularities and use it to classify the phase space as the three circulations vary.
Small-Mass Asymptotics of Massive Point Vortex Dynamics in Bose--Einstein Condensates I: Averaging and Normal Forms
Tomoki Ohsawa, Andrea Richaud, and Roy Goodman. arXiv preprint arXiv:2503.19222, 2026.
We perform an asymptotic analysis of massive point-vortex dynamics in Bose–Einstein condensates in the small-mass limit \(\varepsilon \to 0\). We define two distinguished manifolds in the phase space of the …
QGLAB: A MATLAB Package for Computations on Quantum Graphs
Roy Goodman, Gracie Conte, and Jeremy Marzuola. SIAM Journal on Scientific Computing, 2025.
We describe QGLAB, a new MATLAB package for analyzing partial differential equations on quantum graphs. The supplementary material contains additional examples and a complete listing of user-callable functions.
A new canonical reduction of three-vortex motion and its application to vortex-dipole scattering
Atul Anurag (अतुल अनुराग), Roy Goodman, and Ellison O'Grady. Physics of Fluids, 2024.
We use Jacobi coordinates and Nambu brackets to derive a new representation of the motion of three vortices which, unlike all previous reductions, introduces no singularities into the system. We use this to study the scattering of dipoles by stationary vortices.
An English Translation of Gröbli's Ph.D. Dissertation: "Specielle Probleme über die Bewegung geradliniger paralleler Wirbelfäden"
Roy Goodman. arXiv preprint arXiv:2404.01305, 2024.
A complete English translation of Walter Gröbli’s 1877 Ph.D. Thesis, together with some notes on the process. The work considers the dynamics of point vortices in a two-dimensional inviscid incompressible fluid and derives a number of exact solutions in the cases of three, four and \(2n\) vortices with certain restriction on the vortices’ circulation and the symmetry of the initial configuration.
An Optimal Control Approach to Gradient-Index Design for Beam Reshaping
Jimmie Adriazola and Roy Goodman. Journal of the Optical Society of America A, 2022.
We address the problem of reshaping light in the Schrödinger optics regime from the perspective of optimal control theory. In technological applications, Schrödinger optics is often used to model a …
Recent Posts
How should we LaTeX?
TL;DR: Start from my template on GitHub, then read the style guide below.
Last updated October 2026. I first wrote this post in 2021 around the physics package. That package hasn’t …
Jul 2, 2021 · 14 min read
matdfield & matpplane (2021, now superseded)
Use these instead: MathWorks’ Phase Plane and Slope Field apps. I’m happy to report that my work here has been rendered obsolete. Brian Hong at MathWorks has written new programs …
Feb 2, 2021 · 4 min read
Summary of Responses to my post to the SIAG-DS mailing list
Written in January 2021. Several of the tools below have changed since then, and the best option for MATLAB users is now MathWorks’ Phase Plane and Slope Field apps.
Feb 2, 2021 · 6 min read
Contact
- goodman@njit.edu
- 973 642 4261
- Department of Mathematical Sciences, University Heights, Newark, NJ 07102
- 504A Cullimore Hall. If coming from Newark Penn Station, go to the Newark Light Rail, buy an Underground fare from the machine, validate it, and board at the far track for a train bound for Branch Brook Park or Grove Street. Get out at Warren Street/NJIT.
- Tuesday/Friday 3:00–4:00
- Zoom Room