4 Year GTA - Numerical Methods for Non-Classical Lippmann-Schwinger Equations in Wave Scattering

About the Project

Open to UK Applicants only

Mathematics are offering 3 fully-funded Graduate Teaching Assistant (GTA) PhD studentships available for UK applicants, starting in September 2026.

Graduate Teaching Assistantships allow research students to fund their PhD through part-time teaching work with the University.

A Graduate Teaching Assistant is responsible to the Head of School and is expected to undertake teaching or other duties within the School - not normally exceeding 8-10 contact hours per week - while undertaking research leading to a PhD.

Approximately 80% of their time will be spent on their doctoral research and 20% on their GTA responsibilities. Training is provided to help Graduate Teaching Assistants develop their teaching related skills and enhance their professional competencies.

Project Highlights

• Blend deep analytical insight with modern numerical algorithms to overcome the limitations of current scattering solvers.
• Enable robust simulations of high-contrast materials, sharp geometries, and engineered microstructures—key to acoustics, photonics, and advanced material design.
• Join an active international network of collaborators across Europe and the UK, with opportunities for research visits and joint publications.

Project description

Wave scattering underpins technologies from ultrasound and radar to photonics and metamaterials. The Lippmann–Schwinger (LS) equation, which reformulates wave propagation as an integral equation, is a powerful and well-established analytical and computational tool, highly effective across a broad range of classical scattering problems. But the scenarios that matter most in modern applications often lie outside classical LS theory.

When a material has a discontinuity in the divergence part of the governing equation — a situation encountered in realistic electromagnetic scattering and engineered metamaterials — the LS operator loses the compactness properties on which standard theory relies. The result is a non-classical LS operator that is mathematically more delicate and for which standard numerical methods can become unstable or inaccurate.

Recent analytical work shows that a variational formulation of the non-classical LS operator can be interpreted as a smooth perturbation of a well-understood static problem. This opens a promising route: the static case has rich solution structure, and the key idea of this project is to exploit that structure to design physically informed numerical bases for the full scattering problem.

The PhD student will construct spectral-type schemes in which the scattered field is expanded in basis functions derived from the static solution. These act as a natural coordinate system for the LS equation, enabling stable, efficient solvers in exactly the high-contrast, geometrically complex regimes where standard methods fail.

The project will proceed in stages, from waveguide models, through multi-particle metamaterial configurations, to three-dimensional acoustics, Maxwell's equations, and elasticity. This progression will ensure the student develops deep mathematical understanding alongside modern computational expertise, building a cross-disciplinary skill set relevant to mathematics, modelling in engineering, physics, and materials science.

Project enquiries to Dr Matias Ruiz mr447@le.ac.uk

Application enquiries to pgrapply@le.ac.uk

To apply please refer to the application advice and use the application link at https://le.ac.uk/study/research-degrees/funded-opportunities/maths-gta

Funding Notes

The 4 year GTA funded studentships provide:

• A combined teaching and stipend payment, currently. for 2026/7 this will be £21,805 per year, paid in monthly instalments
• Tuition fees at UK rates