(MSc, PhD)

**Senior LecturerDirector of Studies for 100-level Mathematics**

Office: Science III, room 212

Tel +64 3 479 7783

Email robert.vangorder@otago.ac.nz

## About

I obtained BS (2009) and PhD (2014) degrees in mathematics from the University of Central Florida (UCF), USA, where I held a Trustees Doctoral Fellowship (2009-2011) and then a National Science Foundation Graduate Research Fellowship (2011-2014). After completion of my PhD, I moved to the Mathematical Institute at the University of Oxford, UK, and took up the post of Research Fellow in Nonlinear Dynamics (2014-2015), followed by a Glasstone Research Fellowship in Science (2015-2018). In 2019, I joined the faculty of the Department of Mathematics and Statistics at the University of Otago as a Senior Lecturer.

## Teaching responsibilities

Robert's teaching responsibilities include:

- MATH 140 Fundamentals of Modern Mathematics 2
- MATH 302 Complex Analysis
- MATH4A1 Techniques in Applied Mathematics 1
- MATH4A2 Techniques in Applied Mathematics 2

## Research Interests

In my research I seek to better understand how physical phenomena can be described, predicted, and even modified using tools from applied mathematics. I specialize in the trifecta of mathematical modelling (writing down relations between physical quantities within the language of mathematics... this is where all of those equations come from!), analytical and asymptotic solution methods (deriving an exact or approximate solution to a mathematical problem using pen-and-paper approaches), and numerical simulations (using a computer program to solve a mathematical problem). I apply these methods to study physical phenomena, primarily those falling into one of the following areas:

### Fluid dynamics

If you’re alive, you’re interacting with fluids: The air you breathe and the water you drink are examples of fluids. Despite the ubiquitous nature of fluids, there are still many questions we have about their behavior. My interests in fluids include better understanding the dynamics governing fundamental structures – such as vortices, bubbles, waves, and boundary layers – that can then be used as building blocks for more complex fluid flows. What forms can these structures take? Do they persist, or break apart over time? What happens if we attempt to manipulate or control these structures? These are the kinds of questions I am interested in.

### Spatial instabilities and pattern formation

Diffusive instabilities – such as the Turing and Benjamin-Feir instabilities – have been proposed as mechanisms for the formation of patterns in many real-world systems, ranging from spot formation on the coats of big cats to optical turbulence in lasers. I am interested in understanding how these instability mechanisms extended to more generic non-autonomous or spatially heterogeneous systems, where they result in messier yet perhaps more realistic patterning. These heterogeneous systems arise from models of reaction-diffusion processes such as chemical reactions in the presence of thermal forcing, fluid flows, or evolving space domains, and are found in applications ranging from physics and chemistry to biology and epidemiology. Under what conditions can spatial or spatiotemporal patterns emerge from such systems? How might these patterns be controlled?

### Quantum theory, quantum fluids, and nonlinear waves

My interests in theoretical physics include quantum mechanics and quantum field theory, with a particular focus on modelling problems in low-temperature physics and condensed matter physics as quantum fluids. Some of my work in this area has involved understanding the dynamics Bose-Einstein condensates, quantized vortex filaments in superfluid helium, and confined quantum systems. While these topics are farther removed from our daily experiences, they exhibit exotic and interesting behaviors, providing a number of fascinating scientific challenges. Nonlinear waves arise in these and other applications, and I am interested in understanding how nonlinear waves evolve under various conditions relevant to realistic experimental configurations, such as confined space domains or external forcing.

I am happy to supervise PhD and honours projects in any of these areas.

## Publications

Van Gorder, R. A. (2022). Compressed hydrogen atoms confined within generic boxes.
*Proceedings of the Royal Society A*, *478*(2268), 20220467. doi:
10.1098/rspa.2022.0467

Luckins, E. K., Oliver, J. M., Please, C. P., Sloman, B. M., & Van Gorder, R. A. (2022).
Modelling and analysis of an endothermic reacting counter-current flow. *Journal
of Fluid Mechanics*, *949*, A21. doi:
10.1017/jfm.2022.702

Luckins, E. K., Oliver, J. M., Please, C. P., Sloman, B. M., Valderhaug, A. M., &
Van Gorder, R. A. (2022). Modelling alternating current effects in a submerged arc
furnace. *IMA Journal of Applied Mathematics*, *87*, 492-520. doi:
10.1093/imamat/hxac012

Ritchie, J. S., Krause, A. L., & Van Gorder, R. A. (2022). Turing and wave instabilities
in hyperbolic reaction–diffusion systems: The role of second-order time derivatives
and cross-diffusion terms on pattern formation. *Annals of Physics*. Advance
online publication. doi: 10.1016/j.aop.2022.169033

Van Gorder, R. A. (2022). Adiabatic soliton management: Controlling solitary wave
motion while keeping the wave envelope unchanged. *Physics Letters A*. Advance
online publication. doi:
10.1016/j.physleta.2022.128284

Vajravelu, K., & Van Gorder, R. A. (2012). *Nonlinear flow phenomena and homotopy
analysis: Fluid flow and heat transfer*. Heidelberg, Germany: Springer, 190p.
doi: 10.1007/978-3-642-32102-3

Authored Book - Research

Van Gorder, R. A. (2014). Stability of auxiliary linear operator and convergence-control
parameter in the homotopy analysis method. In S. Liao (Ed.), *Advances in the
homotopy analysis method*. (pp. 123-180). World Scientific. doi:
10.1142/9789814551250_0004

Chapter in Book - Research

Luckins, E. K., Oliver, J. M., Please, C. P., Sloman, B. M., & Van Gorder, R. A. (2022).
Modelling and analysis of an endothermic reacting counter-current flow. *Journal
of Fluid Mechanics*, *949*, A21. doi:
10.1017/jfm.2022.702

Journal - Research Article

Luckins, E. K., Oliver, J. M., Please, C. P., Sloman, B. M., Valderhaug, A. M., &
Van Gorder, R. A. (2022). Modelling alternating current effects in a submerged arc
furnace. *IMA Journal of Applied Mathematics*, *87*, 492-520. doi:
10.1093/imamat/hxac012

Journal - Research Article

Ritchie, J. S., Krause, A. L., & Van Gorder, R. A. (2022). Turing and wave instabilities
in hyperbolic reaction–diffusion systems: The role of second-order time derivatives
and cross-diffusion terms on pattern formation. *Annals of Physics*. Advance
online publication. doi: 10.1016/j.aop.2022.169033

Journal - Research Article

Van Gorder, R. A. (2022). Adiabatic soliton management: Controlling solitary wave
motion while keeping the wave envelope unchanged. *Physics Letters A*. Advance
online publication. doi:
10.1016/j.physleta.2022.128284

Journal - Research Article

Van Gorder, R. A. (2022). Compressed hydrogen atoms confined within generic boxes.
*Proceedings of the Royal Society A*, *478*(2268), 20220467. doi:
10.1098/rspa.2022.0467

Journal - Research Article

Van Gorder, R. A. (2022). Finite time blowup of incompressible flows surrounding compressible
bubbles evolving under soft equations of state. *Proceedings of the Royal Society
A*, *478*(2262), 20220172. doi:
10.1098/rspa.2022.0172

Journal - Research Article

Erban, R., & Van Gorder, R. A. (2021). On the sum of positive divisors functions.
*Research in Number Theory*, *7*, 25. doi:
10.1007/s40993-021-00240-6

Journal - Research Article

Luckins, E. K., Oliver, J. M., Please, C. P., Sloman, B. M., & van Gorder, R. A. (2021).
Homogenised model for the electrical current distribution within a submerged arc furnace
for silicon production. *European Journal of Applied Mathematics*. Advance
online publication. doi: 10.1017/s0956792521000243

Journal - Research Article

Van Gorder, R. A. (2021). A theory of pattern formation for reaction–diffusion systems
on temporal networks. *Proceedings of the Royal Society A*, *477*, 20200753.
doi: 10.1098/rspa.2020.0753

Journal - Research Article

Van Gorder, R. A. (2021). Pattern formation from spatially heterogeneous reaction-diffusion
systems. *Philosophical Transactions of the Royal Society A*, *379*,
20210001. doi: 10.1098/rsta.2021.0001

Journal - Research Article

Van Gorder, R. A. (2021). Time-varying Bose–Einstein condensates. *Proceedings
of the Royal Society A*, *477*, 20210443. doi:
10.1098/rspa.2021.0443

Journal - Research Article

Van Gorder, R. A., Kamilova, A., Birkeland, R. G., & Krause, A. L. (2021). Locating
the baking isotherm in a Søderberg electrode: Analysis of a moving thermistor model.
*SIAM Journal on Applied Mathematics*, *81*(4), 1691-1716. doi:
10.1137/20M1314276

Journal - Research Article

Van Gorder, R. A., Klika, V., & Krause, A. L. (2021). Turing conditions for pattern
forming systems on evolving manifolds. *Journal of Mathematical Biology*, *82*,
4. doi: 10.1007/s00285-021-01552-y

Journal - Research Article

González-Fariña, R., Münch, A., Oliver, J. M., & van Gorder, R. A. (2020). Modeling
microsilica particle formation and growth due to the combustion reaction of silicon
monoxide with oxygen. *SIAM Journal on Applied Mathematics*, *80*(2),
1003-1033. doi: 10.1137/19M1287080

Journal - Research Article

Krause, A. L., & Van Gorder, R. A. (2020). A non-local cross-diffusion model of population
dynamics II: Exact, approximate, and numerical traveling waves in single- and multi-species
populations. *Bulletin of Mathematical Biology*, *82*, 113. doi:
10.1007/s11538-020-00787-y

Journal - Research Article

Sloman, B. M., Please, C. P., & Van Gorder, R. A. (2020). Melting and dripping of
a heated material with temperature-dependent viscosity in a thin vertical tube. *Journal
of Fluid Mechanics*, *905*, A16. doi:
10.1017/jfm.2020.727

Journal - Research Article

Taylor, N. P., Kim, H., Krause, A. L., & Van Gorder, R. A. (2020). A non-local cross-diffusion
model of population dynamics I: Emergent spatial and spatiotemporal patterns. *Bulletin
of Mathematical Biology*, *82*, 112. doi:
10.1007/s11538-020-00786-z

Journal - Research Article

Van Gorder, R. A. (2020). Influence of temperature on Turing pattern formation. *Proceedings
of the Royal Society A*, *476*(2240), 20200356. doi:
10.1098/rspa.2020.0356

Journal - Research Article

Van Gorder, R. A. (2020). Perturbation theory for Bose-Einstein condensates on bounded
space domains. *Proceedings of the Royal Society A*, *476*, 20200674.
doi: 10.1098/rspa.2020.0674

Journal - Research Article

Van Gorder, R. A. (2020). Turing and Benjamin–Feir instability mechanisms in non-autonomous
systems. *Proceedings of the Royal Society A*, *476*(2238). doi:
10.1098/rspa.2020.0003

Journal - Research Article

Van Gorder, R. A., Krause, A. L., Malomed, B. A., & Kaup, D. J. (2020). Unstaggered-staggered
solitons on one- and two-dimensional two-component discrete nonlinear Schrödinger
lattices. *Communications in Nonlinear Science & Numerical Simulation*, *85*,
105244. doi: 10.1016/j.cnsns.2020.105244

Journal - Research Article

Xu, Y., Krause, A. L., & Van Gorder, R. A. (2020). Generalist predator dynamics under
Kolmogorov versus non-Kolmogorov models. *Journal of Theoretical Biology*,
*486*, 110060. doi:
10.1016/j.jtbi.2019.110060

Journal - Research Article

Conti, F., & Van Gorder, R. A. (2019). The role of network structure and time delay
in a metapopulation Wilson-Cowan model. *Journal of Theoretical Biology*, *477*,
1-13. doi: 10.1016/j.jtbi.2019.05.010

Journal - Research Article

Fadai, N. T., Please, C. P., & Van Gorder, R. A. (2019). Modelling structural deformations
in a roasting coffee bean. *International Journal of Non-Linear Mechanics*,
*110*, 123-130. doi:
10.1016/j.ijnonlinmec.2018.12.006

Journal - Research Article

Fussell, E. F., Krause, A. L., & Van Gorder, R. A. (2019). Hybrid approach to modeling
spatial dynamics of systems with generalist predators. *Journal of Theoretical
Biology*, *462*, 26-47. doi:
10.1016/j.jtbi.2018.10.054

Journal - Research Article

Horner, C. L., & Van Gorder, R. A. (2019). Dynamics of quantized vortex filaments
under a local induction approximation with second-order correction. *Physics of
Fluids*, *31*(6), 065103. doi:
10.1063/1.5091567

Journal - Research Article

Hua, T., & Van Gorder, R. A. (2019). Wave propagation and pattern formation in two-dimensional
hexagonally-packed granular crystals under various configurations. *Granular Matter*,
*21*, 3. doi: 10.1007/s10035-018-0852-8

Journal - Research Article

Kiradjiev, K. B., Halvorsen, S. A., Van Gorder, R. A., & Howison, S. D. (2019). Maxwell-type
models for the effective thermal conductivity of a porous material with radiative
transfer in the voids. *International Journal of Thermal Sciences*, *145*,
106009. doi: 10.1016/j.ijthermalsci.2019.106009

Journal - Research Article

Perkins, M., & Van Gorder, R. A. (2019). Closed-form calculation of infinite products
of Glaisher-type related to Dirichlet series. *Ramanujan Journal*, *49*,
371-389. doi: 10.1007/s11139-018-0037-4

Journal - Research Article

Sachak-Patwa, R., Fadai, N. T., & Van Gorder, R. A. (2019). Modeling multi-group dynamics
of related viral videos with delay differential equations. *Physica A*, *521*,
197-217. doi: 10.1016/j.physa.2019.01.052

Journal - Research Article

Van Gorder, R. A. (2019). Optimal homotopy analysis and control of error for implicitly
defined fully nonlinear differential equations. *Numerical Algorithms*, *81*,
181-196. doi: 10.1007/s11075-018-0540-0

Journal - Research Article

Van Gorder, R. A., Kim, H., & Krause, A. L. (2019). Diffusive instabilities and spatial
patterning from the coupling of reaction-diffusion processes with Stokes flow in complex
domains. *Journal of Fluid Mechanics*, *877*, 759-823. doi:
10.1017/jfm.2019.620

Journal - Research Article

Van Gorder, R. A., Krause, A. L., & Kwiecinski, J. A. (2019). Amplitude death criteria
for coupled complex Ginzburg–Landau systems. *Nonlinear Dynamics*, *19*,
151-159. doi: 10.1007/s11071-019-04961-3

Journal - Research Article

Zhou, Z., & Van Gorder, R. A. (2019). Turing instability and colony formation in spatially
extended Rosenzweig-MacArthur predator-prey models with allochthonous resources. *Bulletin
of Mathematical Biology*, *81*, 5009-5053. doi:
10.1007/s11538-019-00667-0

Journal - Research Article

Baines, L. W. S., & Van Gorder, R. A. (2018). Soliton wave-speed management: Slowing,
stopping, or reversing a solitary wave. *Physical Review A*, *97*, 063814.
doi: 10.1103/PhysRevA.97.063814

Journal - Research Article

Bakhtiari, P., Abbasbandy, S., & Van Gorder, R. A. (2018). Reproducing kernel method
for the numerical solution of the 1D Swift–Hohenberg equation. *Applied Mathematics
& Computation*, *339*, 132-143. doi:
10.1016/j.amc.2018.07.006

Journal - Research Article

Bakhtiari, P., Abbasbandy, S., & Van Gorder, R. A. (2018). Solving the Dym initial
value problem in reproducing kernel space. *Numerical Algorithms*, *78*(2),
405-421. doi: 10.1007/s11075-017-0381-2

Journal - Research Article

Bhatt, A., & Van Gorder, R. A. (2018). Chaos in a non-autonomous nonlinear system
describing asymmetric water wheels. *Nonlinear Dynamics*, *93*(4), 1977-1988.
doi: 10.1007/s11071-018-4301-3

Journal - Research Article

Eide, R. M., Krause, A. L., Fadai, N. T., & Van Gorder, R. A. (2018). Predator-prey-subsidy
population dynamics on stepping-stone domains with dispersal delays. *Journal of
Theoretical Biology*, *451*, 19-34. doi:
10.1016/j.jtbi.2018.04.038

Journal - Research Article

Fadai, N. T., Please, C. P., & Van Gorder, R. A. (2018). Asymptotic analysis of a
multiphase drying model motivated by coffee bean roasting. *SIAM Journal on Applied
Mathematics*, *78*(1), 418-436. doi:
10.1137/16M1095500

Journal - Research Article

Flora Meng, X., Van Gorder, R. A., & Porter, M. A. (2018). Opinion formation and distribution
in a bounded-confidence model on various networks. *Physical Review E: Statistical,
Nonlinear, & Soft Matter Physics*, *97*(2), 022312. doi:
10.1103/PhysRevE.97.022312

Journal - Research Article

Jansen, J. E., & Van Gorder, R. A. (2018). Dynamics from a predator-prey-quarry-resource-scavenger
model. *Theoretical Ecology*, *11*(1), 19-38. doi:
10.1007/s12080-017-0346-z

Journal - Research Article

Kekić, A., & Van Gorder, R. A. (2018). Wave propagation across interfaces induced
by different interaction exponents in ordered and disordered Hertz-like granular chains.
*Physica D*, *384-385*, 18-33. doi:
10.1016/j.physd.2018.07.007

Journal - Research Article

Kennedy, D. T., & Van Gorder, R. A. (2018). Motion of open vortex-current filaments
under the Biot-Savart model. *Journal of Fluid Mechanics*, *836*, 532-559.
doi: 10.1017/jfm.2017.826

Journal - Research Article

Kleanthous, A., Hua, T., Manai, A., Yawar, K., & Van Gorder, R. A. (2018). Dynamics
from a mathematical model of a two-state gas laser. *Physica A*, *497*,
26-40. doi: 10.1016/j.physa.2017.12.110

Journal - Research Article

Krause, A. L., Beliaev, D., Van Gorder, R. A., & Waters, S. L. (2018). Bifurcations
and dynamics emergent from lattice and continuum models of bioactive porous media.
*International Journal of Bifurcation & Chaos*, *28*(11), 1830037.
doi: 10.1142/S0218127418300379

Journal - Research Article

Krause, A. L., Burton, A. M., Fadai, N. T., & Van Gorder, R. A. (2018). Emergent structures
in reaction-advection-diffusion systems on a sphere. *Physical Review E: Statistical,
Nonlinear, & Soft Matter Physics*, *97*(4), 042215. doi:
10.1103/PhysRevE.97.042215

Journal - Research Article

Krause, A. L., Kurowski, L., Yawar, K., & Van Gorder, R. A. (2018). Stochastic epidemic
metapopulation models on networks: SIS dynamics and control strategies. *Journal
of Theoretical Biology*, *449*, 35-52. doi:
10.1016/j.jtbi.2018.04.023

Journal - Research Article