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Nonlinear waves on graphs

   School of Mathematics

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  Dr D Proment  No more applications being accepted  Self-Funded PhD Students Only

About the Project

Waves can be found in almost every physical system. When the wave amplitude is sufficiently large, nonlinear effects need to be considered: physicists call such waves “nonlinear waves”. Unveiling the structure and dynamics of nonlinear waves is a notoriously difficult problem.

This research project aims at studying discrete nonlinear waves on graphs. An example of graph where waves propagate is the neural brain network, which can be thought as an extremely complicate “circuit” of connections where electromagnetic-type waves propagate; other wave-like structures moving on graphs are the spreading of information on social networks, the flow in systems like the human circulatory system, pipe networks, and porous media. Different discrete versions of nonlinear wave models like the Duffin equation, the nonlinear Schroedinger equation, the Korteweg-de Vrie equation and other inherently discrete systems like the Fermi-Pasta-Ulam-Tsingou model will be considered. The goal is to characterise the nonlinear waves in each model depending on the underlying graph, making use of the graph’s spectral properties, dimension, and topology.

This project is by nature interdisciplinary: it involves mixing branches of physics (nonlinear physics, Hamiltonian and Lagrangian systems, statistical mechanics), applied mathematics (nonlinear ODEs and PDEs, eigenmode and eigenfunction decomposition), and pure mathematics (graph theory, group theory, differential geometry). Due to the complexity of the proposed research, it is likely that much of the work will be carried using extensive numerical simulations. Possible links with experimentalists will be considered throughout the project.

We are looking for an enthusiastic, talented, and extremely motivated applicant with a degree in Physics and/or Mathematics (or similar). Highly desirable skills are knowledge of nonlinear waves and graph theory, and excellent ability in numerical coding.

For more information on the project details and the research environment, please contact Dr Davide Proment ([Email Address Removed]).

Funding Notes

This PhD project is offered on a self-funding basis. It is open to applicants with funding or those applying to funding sources. Details of tuition fees can be found at
A bench fee is also payable on top of the tuition fee to cover specialist equipment or laboratory costs required for the research. Applicants should contact the primary supervisor for further information about the fee associated with the project.


i) Onorato, M., Vozella, L., Proment, D., & Lvov, Y. V. (2015). Route to thermalization in the α-Fermi–Pasta–Ulam system. Proceedings of the National Academy of Sciences, 112(14), 4208-4213.
ii) Caputo, J. G., Khames, I., Knippel, A., & Aceves, A. B. (2018). Localized solutions of nonlinear network wave equations. Journal of Physics A: Mathematical and Theoretical, 52(3), 035101.
iii) Porter, M. A. (2020). Nonlinearity+ networks: A 2020 vision. In Emerging Frontiers in Nonlinear Science (pp. 131-159). Springer, Cham.
iv) Kollár, A. J., Fitzpatrick, M., & Houck, A. A. (2019). Hyperbolic lattices in circuit quantum electrodynamics. Nature, 571(7763), 45-50.
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