Discrete and continuous integrable systems (Ref: MV/MA/2024)

Continuous integrable systems are nonlinear differential equations that can be solved exactly and exhibit orderly dynamics. Integrability is the result of some structure underlying the differential equation. A wide variety of such structures exists, formulated through, for example, symmetry groups, differential geometry, or spectral theory. Discrete integrable systems are difference equations that exhibit similar behaviour and possess similar structures to continuous integrable systems. Despite this similarity, it is a non-trivial problem to find explicit relations between discrete and continuous integrable systems.

The goal of this project is to develop connections between discrete and continuous integrable systems, particularly those for which the underlying structure is a “Lax pair”. A Lax pair can be thought of as a system of linear equations, which are compatible if and only if an additional condition is satisfied. If we can write a given nonlinear differential (or difference) equation as the compatibility condition of a Lax pair, that provides it with sufficient structure to make it an integrable system. In this project we will explore algorithmic approaches to turn discrete Lax pairs into continuous ones.

Integrable systems provide valuable models in fundamental physics, signal processing, wave dynamics, and other fields. The relations between continuous and discrete integrable systems also provide useful insights towards efficient numerical approximation of broader classes of differential equations.

The successful candidate will be part of the Geometry and Mathematical Physics group at Loughborough University, benefitting from a stimulating environment that includes weekly research seminars, a diverse expertise in integrable systems, geometry, and dynamics, as well as links with research groups across the UK and internationally. They will be supported in developing their research and science communication skills, and will be able to partake in outreach events if desired. They will have the opportunity to advance their coding skills and learn new programming languages. Loughborough University provides supportive and flexible working arrangements. It is a member of the Race Equality Charter, a Disability Confident Employer, and a Stonewall Diversity Champion. The School of Science holds an Athena SWAN bronze award for gender equality.

Entry requirements:

Students should have, or expect to achieve, at least a 2:1 honours (or international equivalent) in Mathematics or related subjects. A relevant master’s degree and/or a background in mathematical physics and/or familiarity with symbolic computations (Mathematica, SageMath, or similar) will be desirable.

English language requirements:

Applicants must meet the minimum English language requirements. Further details are available on the International website.

How to apply:

All applications should be made online. Under programme name, select Mathematical Sciences. Please quote the advertised reference number: MV/MA/2024 in your application.

To avoid delays in processing your application, please ensure that you submit the minimum supporting documents, including a CV.

The selection criteria will be used by academic schools to help them make a decision on your application.

Project search terms:

mathematics, theoretical physics, mathematical physics, integrable systems

Email Address Sci:

sci-pgr@lboro.ac.uk