I am a Georgina Sweet Australian Laureate Fellow in mathematics at the University of Sydney. I develop mathematical methods to study solutions of integrable systems, which arise as universal models in physics, such as the Painlevé equations.
My research interests lie in non-linear differential and difference equations, with a particular focus on asymptotic methods. Currently, I am creating a geometric framework to reveal properties of critical solutions of nonlinear models that reflect universal structures in physical models.
Nonlinear integrable systems arise in a wide variety of fields, including ion channel transport in biology, random matrix theory, string theory and orthogonal polynomial theory. Many such systems appear as universal mathematical models in certain contexts. In the case of one continuous dimension, the systems of interest are called the Painlevé equations, while in two or more dimensions, the equations are soliton equations.
Their discrete versions have been at the leading edge of developments in recent times. However, the emphasis in the field has been focused on the identification of integrable discrete versions of the Painlevé equations and soliton equations. These equations provide an insight into physical models, such as random matrix theory. However, our knowledge of their solutions remains extremely limited.
My specific research areas include:
- Integrable systems
- Painlevé equations
- Geometric asymptotics
- Nonlinear dynamics
- Nonlinear waves
- Perturbation theory
Research, publications and travel
nalini.joshi [at] sydney.edu.au