# Integrable systems

### Summary

The field of integrable systems is relatively young but has stimulated great interest amongst physicists (in the theory of random matrices, string theory, or quantum gravity) and mathematicians (in the theory of orthogonal polynomials, Nevanlinna theory, geometry and soliton theory). Many integrable systems appear universally as mathematical models. Integrable equations can be differential equations or difference equations. The area has led to outstanding results but many problems remain open.

### Supervisor(s)

### Research Location

School of Mathematics and Statistics

### Program Type

PHD

### Synopsis

While the initial value problem has received enormous attention for integrable differential equations, hardly any attention has been given to corresponding initial value problems for crucial classes of integrable two-dimensional discrete equations called lattice equations. The simplest, symmetric integrable lattice equations also possess surprising properties such as multi-dimensional consistency in three dimensional space. Do we really know all the possible multi-dimensionally consistent lattice equations? How do these relate to other integrable ordinary difference equations? These are challenging but beautiful problems. The aim of this project is to answer one of these problems.

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### Keywords

Integrable systems, orthogonal polynomials, soliton theory, differential equations, difference equations, Painleve equations, lattice equations, inverse scattering transform, consistency around a cube, symmetry reductions

### Opportunity ID

The opportunity ID for this research opportunity is: **636**

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