Geometry and Asymptotics of 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

There are a number of major gaps in the theory of integrable systems, which have prevented us from finding properties that are highly sought after in applications. For example, despite our knowledge of special families of exact solutions, we do not know how general solutions of discrete Painlevé equations behave anywhere in the domain of the independent variable. Despite the importance of bounded real solutions in applications, we do not know the locations of multiple poles or zeroes of these solutions. This program will advance knowledge in asymptotic behaviour and analytic nature of solutions, the existence of applicable solutions that remain bounded on certain intervals, and the interrelationships between the properties of integrable systems. The information we derive will be relevant to mathematicians, physicists, fluid dynamicists and biologists interested in modelling in environments with specified heterogeneities.

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

Painleve equations, discrete Painlevé equations, asymptotics, affine Weyl groups, differential equations, difference equations, soliton theory, Integrable systems

Opportunity ID

The opportunity ID for this research opportunity is: 1146

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