# 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)

### 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.

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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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