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Nonlinear Phenomena
"...the progress of physics will to a large extent depend on the progress of nonlinear mathematics, of methods to solve nonlinear equations..."
W. Heisenberg, Nobel Laureate 1932
Nonlinear equations describe fundamental physical phenomena in nature ranging from chaotic behaviour in biological systems, plasma containment in tokamaks and stellarators for energy generation, to solitonic fibre optical communication devices. The Nonlinear Phenomena Group at UNSW is world-renowned for its work in soliton theory and dynamical systems and attracts visitors of international repute in these areas on a regular basis. Its activities have been recognised by Chief Investigator representation in the recently awarded Centre of Excellence for Mathematics and Statistics of Complex Systems. This will provide a base for a major research programme involving the analysis of complex physical and biological systems and the systematic investigation of potential new areas of application of modern soliton theory in magneto-hydrodynamics, the fabrication of fibre-reinforced composites and elastic shell structure design. Research Interests
- solutions to nonlinear systems (approximation, numerical and exact methods)
- fluid mechanics and differential equations
- nonlinear hydrodynamic stability
- nonlinear dynamics and stability
- nonlinear lattice dynamics and non-equilibrium growth models
- superintergrable systems
- differential geometry and symmetries of wave equations
- solar modelling
- elliptic partial differential equations in mathematical, physical and biological sciences
- geometric structure of soliton systems both discrete and continuous
- infinitesimal and finite Bäcklund transformations
- hidden solitonic structures in nonlinear continuum mechanics
- reciprocal transformations and moving boundary value problems
- general relativity
- dynamic equations on time scales
- Lie-algebraic study of symmetries of nonlinear systems of physical interest
- discrete integrable dynamical systems
- symmetry and time-reversal symmetry in dynamical systems
- dynamical systems arising from quasiperiodic physical phenomena
- algebraic dynamics (group theoretic/number theoretic approaches to dynamical systems)
- inequalities - differential, difference, dynamic and integral
- differential inclusions
- transfer operators and Perron-Frobenius operators
- smooth and numerical ergodic theory
Group Members
Visiting Professors
- Mark Ablowitz, Pulse dynamics in nonlinear optics; chaotic dynamics of water waves; solutions of multidimensional nonlinear equations; solutions of novel systems of nonlinear differential equations in terms of automorphic functions
- Andrew Bassom, Fluid mechanics and differential equations, especially the Painlevé equations; nonlinear Schrödinger equation and nonlinear localized buckling phenomena
- Roger Grimshaw, Flow over topography; large amplitude waves; coastal ocean dynamics; internal gravity waves; vortex dynamics; exponential asymptotics
- Pavel Winternitz, Lie-algebraic study of symmetries of nonlinear systems of physical interest
Research Fellows
Postgraduate Students
- Ognjen Stancevic
- Dalia Terhesiu
- Jim Pettigrew, Algebraic dynamics
Collaborators
- Professor W.J Beyn, University of Bielefeld, Germany
- Professor M. Baake, University of Bielefeld, Germany
- Professor M. Batchelor, Australian National University
- Professor Dr. Michael Dellnitz University of Paderborn, Germany
- Professor Dr. Oliver Junge TU Munich, Germany
- Professor Christopher Bose University of Victoria, Canada
- Professor Anthony Quas University of Victoria, Canada
- Dr. I. Benn, University of Newcastle, Australia
- Professor A.I. Bobenko, Technical University Berlin, Germany
- Professor F. Burstall, University of Bath, UK
- Professor H. W. Capel, University of Amsterdam, The Netherlands
- Professor E. Kalnins, University of Waikato, New Zealand
- Dr. A.D. King, University of Bath, UK
- Dr. M. Kléman, Université Pierre-et-Marie-Curie, Paris, France
- Professor B.G. Konopelchenko, Università di Lecce, Italy
- Dr J. Lamb, Imperial College, UK
- Professor W. Miller, Jr., University of Minnesota, USA
- Professor R. Quispel, La Trobe University, Australia
- Professor F. Vivaldi, University of London, UK
- Dr. S. Wearne, Mount Sinai, USA
Relevant Courses
- MATH2260 Dynamical Systems - 2nd year
- MATH3041 Mathematical Modelling for Real World Systems - 3rd year
- MATH3201 Nonlinear Systems and Chaos - 3rd year
- MATH5130 Advanced Mathematical Methods - graduate level
- MATH5205 Nonlinear Analysis (Solitons) - graduate level
- MATH5215 Topics in Dynamics (Chaos) - graduate level
Links
Related Groups
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