## Mathematical Physics

Researchers: Cormac Breen; Rossen Ivanov; Stephen O'Sullivan; Emil Prodanov

### Description of area

Integrable systems & solitons (Rossen Ivanov)

In many real-world phenomena nonlinear effects play an essential role. This has motivated an enormous amount of research into the study of nonlinearity, especially in the context of nonlinear waves (see also our Fluid Mechanics research page). The interplay between nonlinearity and other characteristics of a model can be illustrated with relatively simple model equations. Perhaps the best known examples of integrable equations arising in nonlinear theories are the KdV, sin-Gordon and the Nonlinear Schroedinger (NLS) equations. For example, the Korteweg-de Vries (KdV) equation is a well-known equation describing the competition between nonlinear and dissipative effects in water waves.

The KdV equation is an example of an integrable nonlinear PDE which has solutions with special properties, known as solitary waves or solitons. In recent years two other integrable equations, the Camassa-Holm (CH) equation and the Degasperis-Procesi (DP) equation have been of significant interest to mathematicians across a wide range of areas of expertise: spectral problems, integrable systems, analysis, differential geometry etc.

Our research is on the development of solution methods for the CH, DP equations, their multicomponent generalisations and similar types of equations in various regimes, e.g. the inverse scattering method and the dressing method. Our studies consider the Hamiltonian structures and Hamiltonian hierarchies of the investigated models as well as soliton interactions and possible Fluid Mechanics applications: e.g. shallow and deep water waves, blood and biofluid flows.

Lagrangian & Hamiltonian mechanics (Rossen Ivanov)

Certain equations with hydrodynamic relevance such as KdV, the Camassa-Holm and the Hunter-Saxton equations have an interpretation as geodesic flow equations on the group of diffeomorphisms, preserving certain right-invariant norms. These equations have a Lagrangian given by the norm of a vector field. There is an analogy with the Euler equations in hydrodynamics and the rigid body equations with a fixed point, which are also geodesic equations.

We study generalised geodesic equations, in the case when the Langrangian is left or right-invariant under a certain Lie group G. The Lagrangian, as well as the associated Lie-Poisson structures of the arising models are determined by the corresponding Lie algebra. In this class of models are the so-called G-strand equations that have been introduced in several recent publications.

Quantum, classical & topological field theory; general relativity & cosmology (Emil Prodanov, Rossen Ivanov)

The research related to the study of dynamical systems in cosmology considers the time-evolution dynamics of nonlinear cosmological real gas models. The methods used are from the theory of Hamiltonian dynamical systems and the framework is that of FRWL cosmologies. The cosmological variables used are the expansion rate, given by the Hubble parameter, the energy density, and the temperature. The existence of global first integrals, as well as special (second) integrals, is studied. In addition, the issue of identifying a global first integral as a Hamiltonian (for two-dimensional dynamical system) is addressed.

Quantum field theory in curved spacetimes (Cormac Breen)

Quantum gravity remains one of the most important outstanding problems in physics. Without a full theory, one must rely on approximations. One particularly important approximation is semi-classical gravity, which is the treatment of quantum fields interacting with a classical spacetime metric via the semi-classical Einstein equations. The source term in these equations is the expectation value of the stress-tensor of the quantum fields being considered. The calculation of the expectation value of the stress-tensor in a given spacetime can be a long and complicated process, even for the case of a scalar field. Current research is concerned with the development of methods to facilitate these calculations in spacetimes where heretofore such results have proven elusive (for example the Kerr black hole spacetime or black holes in higher dimensional spacetimes).

High-energy astrophysics (Stephen O'Sullivan)

Massive black holes at the centres of galaxies can act as engines for powerful jets spanning millions of light years. These jets slam into intergalactic gas producing brightly emitting shocks, known as hot-spots, through which jet and background material is dumped into vast radio lobes extended around the jets. These radio lobes are luminous because electrons accelerated to close to the speed of light spiral around magnetic field lines and emit synchrotron radiation. Understanding the mechanisms by which this acceleration may occur is a central problem in astrophysics.

Star formation (Stephen O'Sullivan)

Molecular clouds are often described as stellar nurseries and understanding the dynamics of these vast astronomical bodies is crucial to our understanding of star formation. Of central importance is resolving the question of what roles turbulence and magnetic fields play in these clouds, at scales ranging from the cloud itself, in providing support against gravitational collapse, to the disks around forming stars where angular momentum is released via the ejection of highly collimated jets.

For more information about any of these areas and to discuss opportunities for research please contact the individuals above.