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)

Professor Nalini Joshi

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.

Additional Information

HDR Inherent Requirements

In addition to the academic requirements set out in the Science Postgraduate Handbook, you may be required to satisfy a number of inherent requirements to complete this degree. Example of inherent requirement may include:

- Confidential disclosure and registration of a disability that may hinder your performance in your degree;
- Confidential disclosure of a pre-existing or current medical condition that may hinder your performance in your degree (e.g. heart disease, pace-maker, significant immune suppression, diabetes, vertigo, etc.);
- Ability to perform independently and/or with minimal supervision;
- Ability to undertake certain physical tasks (e.g. heavy lifting);
- Ability to undertake observatory, sensory and communication tasks;
- Ability to spend time at remote sites (e.g. One Tree Island, Narrabri and Camden);
- Ability to work in confined spaces or at heights;
- Ability to operate heavy machinery (e.g. farming equipment);
- Hold or acquire an Australian driver’s licence;
- Hold a current scuba diving license;
- Hold a current Working with Children Check;
- Meet initial and ongoing immunisation requirements (e.g. Q-Fever, Vaccinia virus, Hepatitis, etc.)

You must consult with your nominated supervisor regarding any identified inherent requirements before completing your application.

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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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