## Postgraduate research opportunity

# Elliptic Discrete Integrable Systems

**Number of awards**: 1**Deadline**: Ongoing

### Supervisor(s)

Contact Professor Frank Nijhoff to discuss this project further informally.

### Project description

The term "Integrable Systems" refers to a wide class of very special models, described by nonlinear differential (in the "continuous" case) or difference (in the "discrete" case) equations possessing a number of remarkable properties. One of the outstanding features is that these equations are exactly solvable in the sense that, rather than having to rely on numerical techniques or approximations, these equations allow for exact (albeit highly nontrivial) methods for their solution. Examples of these are the well-known soliton solutions of certain partial differential equations in this class. In the discrete case, the theory behind these model difference equations has been steadily developing, mostly from the early 1990s onward, together with the mathematical theories which had to be developed alongside (as they were largely non-existent in the discrete case).

The project focuses on "elliptic" integrable systems, which are those cases where coefficients and generating quantities for these equations are given in terms of elliptic functions. The latter generalizations of trigonometric functions have a rich mathematical structure, some aspects of which are still being explored (e.g. they play a role in Fermat's last theorem), and the integrable models which are defined through them are in a sense "at the top of the food chain" of models: they form the richest and most general class of equations. To a large extent the solution structure of those elliptic models has yet to be unraveled, and that will form the core of the project. In the project the student will investigate specific examples of such elliptic discrete integrable systems, which will entail not only to try and apply some well-tested techniques to these more complex cases for inding explicit solutions, but also to develop some methods for generating novel examples of such systems. As motivation, these models are expected to have relevance not only for creating novel mathematics, but also potentially for finding new models of fundamental physics.

The project is embedded in the activities of a wider research group in Integrable Systems within the School of Mathematics, comprising several permanent staff, postdocs and postgraduate students. The group runs its own weekly seminar, and intertains close connections with other research groups in the School, e.g. in Algebra, Geometry and Analysis, as well as with the Quantum Information group in Physics.

### Entry requirements

Applicants should have, or expect to obtain, a minimum of a UK upper second class honours degree in Mathematics or a related discipline, or equivalent. Familiarity with complex function theory would be beneficial. The project will employ techniques from a variety of subjects in mathematics (such as differential geometry, analysis and algebra) but most of those techniques can be mastered by students with a solid background in `standard' pure and applied mathematics.

### How to apply

Formal applications for research degree study should be made online through the university's website. Please state clearly in the research information second that the PhD you wish to be considered for is 'Elliptic Discrete Integrable Systems' as well as Professor Frank Nijhoff as your proposed supervisor.

If English is not your first language, you must provide evidence that you meet the University's minimum English Language requirements.

*We welcome scholarship applications from all suitably-qualified candidates, but UK black and minority ethnic (BME) researchers are currently under-represented in our Postgraduate Research community, and we would therefore particularly encourage applications from UK BME candidates. All scholarships will be awarded on the basis of merit.*