Project Description

About the Project

Time-modulated media, whose material properties change rapidly in time, are opening up new ways to control waves. They make possible frequency conversion, parametric amplification, non-reciprocal transport and “temporal crystals” with momentum band gaps. Recent work has shown that modulated Fourier expansions, originally developed to analyse highly oscillatory differential equations, open up exciting opportunities for describing waves propagating through such media. It separates the fast oscillations caused by the modulation from the slowly varying wave dynamics.

This PhD project will extend that framework in several directions:

• Nonlinear waves: analysing wave equations with nonlinear (e.g. Kerr-type) responses in time-modulated media, including energy exchange between harmonics, long-time stability and the emergence of resonances.
• Oscillatory potentials: studying Schrödinger- and Klein-Gordon-type equations with rapidly oscillating, time-dependent potentials. This includes rigorous error bounds for the expansion and the identification of effective (homogenised) dynamics.
• Time-varying metamaterials: applying the theory to high-contrast and subwavelength resonator systems with temporally modulated parameters. The aim is to understand phenomena such as parametric instabilities, topological effects and wave localisation.
• Numerical methods: developing and analysing structure-preserving, multiscale integrators that exploit the modulated Fourier structure for efficient long-time simulation.

The project combines asymptotic and functional analysis with scientific computing. There is scope to collaborate with researchers working on metamaterial design and wave physics.

Funding Information

Funding Notes

Funding may be available through a college or EPSRC scholarship in competition with all other PhD applications.

The scholarship will cover tuition fees, training support, and a stipend at standard rates for 3.5 years.

Strong self-funded applicants worldwide will also be considered.

Eligibility Requirements

We are looking for a highly enthusiastic and motivated student with

• a strong 1st class degree in Mathematics at least BSci level, ideally MMath/MSci/MSc level, or equivalent;
• a solid background in numerical analysis of differential equations;
• excellent programming skills;
• excellent communication skills (oral and written).

Application Process

Informal inquiries should be directed to Dr Jörg Nick, e-mail: [email protected]

Early application is strongly recommended; there is an annual deadline (typically at the end of January) for scholarship applications; however, later applications may also be considered. Applications of overseas candidates can only be considered until the beginning of September.

Strong candidates are encouraged to make an informal inquiry.

References

References

https://www.esaim-m2an.org/articles/m2an/pdf/2026/05/m2an260050.pdf