Dynamic Equations on Measure Chains. Boundary value problems (BVPs) on ``measure chains''are new and useful mathematical equations that describe the world around. This project aims to answer some imporotant and fundamental mathematical questions such as
(i) Under what conditions do BVPs on measure chains actually have solutions?
(ii) If solutions do exist, then what are their properties?
The approach is to use modern tools from mathematical analysis, including topological transversality ....Dynamic Equations on Measure Chains. Boundary value problems (BVPs) on ``measure chains''are new and useful mathematical equations that describe the world around. This project aims to answer some imporotant and fundamental mathematical questions such as
(i) Under what conditions do BVPs on measure chains actually have solutions?
(ii) If solutions do exist, then what are their properties?
The approach is to use modern tools from mathematical analysis, including topological transversality and Leray-Schauder degree.
The project outcomes will
(a) significantly advance current mathematical theory for BVPs on measure chains
(b) unify the theory of BVPs for differential and difference equations
(c) potentially apply to many real-world phenomena.Read moreRead less
Singularities And Classifications Of Integrable Systems. What mathematical models of engineering and nature exclude chaos and have globally predictable solutions? What models occur ubquitously in fields as diverse as photonics and quantum gravity? The answers lie in the theory of integrable systems. We aim to develop powerful new algorithms for identifying integrable models and for deducing their remarkable properties. These algorithms are expected to answer fundamental questions of contempora ....Singularities And Classifications Of Integrable Systems. What mathematical models of engineering and nature exclude chaos and have globally predictable solutions? What models occur ubquitously in fields as diverse as photonics and quantum gravity? The answers lie in the theory of integrable systems. We aim to develop powerful new algorithms for identifying integrable models and for deducing their remarkable properties. These algorithms are expected to answer fundamental questions of contemporary importance. Longer term possible outcomes include applications to nonlinear optics and quantum computing.Read moreRead less
Finite Morse index solutions of nonlinear partial differential equations. We aim to produce mathematics which is of not only of interest to mathematicians but is useful in the study of many physical and biological processes. They occur in the study of processes in industry and the study of the environment.
Transitions and singular behaviour in nonlinear partial differential equations. This research produces crucial mathematical methods and results for the understanding of several important nonlinear problems in applied sciences. The mathematical theory itself is of significant value in nonlinear partial differential equations. The project strengthens Australian research in nonlinear science, a critical area of modern international research, and provides high level training for the next generation ....Transitions and singular behaviour in nonlinear partial differential equations. This research produces crucial mathematical methods and results for the understanding of several important nonlinear problems in applied sciences. The mathematical theory itself is of significant value in nonlinear partial differential equations. The project strengthens Australian research in nonlinear science, a critical area of modern international research, and provides high level training for the next generation of Australian mathematicians.Read moreRead less
Integrable Lattice Equations. When mathematical models are simulated on a computer, the result is a system of partial difference equations, whose solutions evolve with discrete steps on a lattice in space and time. While many tools have been developed to study continuous equations, very few mathematical techniques are available for analysing non-linear lattice equations. We aim to develop techniques of solving the initial-value problem for a class of such equations. Our examples include integrab ....Integrable Lattice Equations. When mathematical models are simulated on a computer, the result is a system of partial difference equations, whose solutions evolve with discrete steps on a lattice in space and time. While many tools have been developed to study continuous equations, very few mathematical techniques are available for analysing non-linear lattice equations. We aim to develop techniques of solving the initial-value problem for a class of such equations. Our examples include integrable lattice equations that arise in the simulation of many physical problems ranging from the progression of shallow water waves to signals in an optical fibre.Read moreRead less
Comparing Einstein to Newton: a mathematical foundation for the Newtonian limit and post-Newtonian expansions. This proposal will benefit the nation in the following ways: (i) to make Australia a world leader in post-Newtonian research, (ii) to contribute to Australia's existing commitment to the search for gravitational waves by providing theoretical tools that will aid in the analysis of gravitational wave data, (iii) to train the next generation of Australian gravitational researchers in a fi ....Comparing Einstein to Newton: a mathematical foundation for the Newtonian limit and post-Newtonian expansions. This proposal will benefit the nation in the following ways: (i) to make Australia a world leader in post-Newtonian research, (ii) to contribute to Australia's existing commitment to the search for gravitational waves by providing theoretical tools that will aid in the analysis of gravitational wave data, (iii) to train the next generation of Australian gravitational researchers in a field whose importance will only grow as the field of gravitational wave astronomy matures, and (iv) to facilitate visits by my collaborators to Australia who will bring world class expertise for the benefit of both Australian students and experts in general relativity.Read moreRead less
Abstract methods for nonlinear partial differential equations. To use abstract methods to study nonlinear partial differential equations where nonlinear effects dominate and where the diffusion is possibly small. These equations arise in many applications of mathematics such as population models and catalysis theory.
Blow-up phenomena in semilinear elliptic partial differential equations. This project will address important questions in semilinear elliptic partial differential equations. It will determine the existence, multiplicity, breaking of symmetry, and concentration phenomena of
solutions of semilinear partial differential equations. The asymptotic behaviour of solutions in semilinear equations has enormous potential to impact on fields in which differential equations can be applied to simple mechani ....Blow-up phenomena in semilinear elliptic partial differential equations. This project will address important questions in semilinear elliptic partial differential equations. It will determine the existence, multiplicity, breaking of symmetry, and concentration phenomena of
solutions of semilinear partial differential equations. The asymptotic behaviour of solutions in semilinear equations has enormous potential to impact on fields in which differential equations can be applied to simple mechanical systems, and built and diverse natural phenomena such as suspension bridges, galaxies and mathematical biology problems.Read moreRead less
Boundary Value Problems for Differential Inclusions. Boundary value problems (BVPs) for differential inclusions are mathematical equations that accurately describe the complex world around us. This project aims to answer important mathematical questions such as:
(i) Under what conditions do BVPs for differential inclusions actually have solutions?
(ii) If solutions do exist, what are their properties?
(iii) If solutions are too complicated to be found explicitly, then how can they be approxim ....Boundary Value Problems for Differential Inclusions. Boundary value problems (BVPs) for differential inclusions are mathematical equations that accurately describe the complex world around us. This project aims to answer important mathematical questions such as:
(i) Under what conditions do BVPs for differential inclusions actually have solutions?
(ii) If solutions do exist, what are their properties?
(iii) If solutions are too complicated to be found explicitly, then how can they be approximated?
The approach is to use modern tools from mathematical analysis, including new differential inequalities.
The project outcomes will:
(a) Significantly advance mathematical knowledge for differential inclusions
(b) Have many applications to areas of science, engineering and technology.Read moreRead less
STABLE AND PEAK SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS. We aim to produce mathematics which is not only of interest to mathematicians but is useful in the study of many physical and biological processes. They occur in processes in industry and the study of the environment.