# Nonlinear Analysis

## Research Interests of the Nonlinear Analysis Group

This group is concerned with the study of nonlinear equations (including nonlinear ordinary and partial differential equations) and their application to a wide variety of problems. The importance of nonlinearity is emphasised by the fact that even very simple nonlinear systems can behave in a complicated way, for example, they may display chaotic behaviour. The field is a very active one at present, with many developments occurring both on the theoretical and the application sides and research of both types occurs within the group.

## Research Areas

**Complete Integrability**

Completely integrable partial differential equations, singularity analysis, Painlevé equations.**Elliptic Partial Differential Equations**(linear and nonlinear)

Effect of the underlying domain, isoperimetric inequalities, study of large solutions or solutions when the diffusion is small (and relation with finite Morse index solutions), topological methods.**Mathematical Biosciences**

The construction of mathematical models of biological systems is an important and rapidly expanding area of research. Such systems are very complex, but mathematics can serve to elucidate underlying principles and identify directions for experimental research. A strength of the biology group is the close contacts with experimental groups at Sydney; a very successful collaboration with the Department of Physiology has been in place for over 10 years, leading to the award of competitive research grants totalling over 1.25 million dollars. Current research topics include: social insects, biological pattern formation, communication in neurons and neural systems, synaptic function, autonomic nervous system, phylogenetics, biomedical ultrasound.

**Researchers:** C. Cosgrove, N. Dancer, D. Daners,
W. Gibson, L. Farnell,
G. Gottwald, N. Joshi, M. Myerscough, S. Santra, K. Wang.

## Contact Person

Prof EN Dancer (N.Dancer@maths.usyd.edu.au)