A general theme is mathematical modelling: finding simple, low dimensional descriptions of complicated (dynamic) phenomena that lead to a better understanding and that may also help to develop simulation tools based on pseudo-analytical, computer algebraic and small-scale numerical programmes.
The analytical methods and ideas are used for problems that appear in the physical and technical sciences. The main application domains are fluid dynamics (surface water waves) and mathematical optics (integrated optical devices).
The development of new mathematical methods is stimulated by executing the research in projects with scientists from other disciplines at universities and industry. In that way we identify generic problems by looking for common features in problems of various origin that look tracktable when appropriate advanced mathematical modelling and analysing techniques are used. Most of the problems are modelled by non-linear and/or nonhomogeneous partial differential equations.
In the modelling of problems from fluid dynamics and optics, the mathematical aspect of the modelling shows itself in the link between physical properties (like conservation of energy, or presence of symmetries) and a priori known properties or structures of classes of model-equations (like Hamiltonian systems).
The basic mathematical methods originate from dynamical system theory, variational methods, and methods for free boundary problems. Occasionally, numerical methods are used to design simulation tools that have the same linkage between essential physical properties and properties of the numerical programme.
Beginning 2002, some 10 PhD-students are involved in the execution of the research; all PhD-positions are externally funded (NWO, STW, EC, KNAW, MARIN).
Research in mathematical optics
concerns for the major part the understanding of devices in
integrated optics, in particular micro-resonators. These are small disks or
squares, placed between parallel waveguides, that transfer one colour of light,
entering through one waveguide to the other waveguide, while all other colours
remain in the same waveguide. This
‘filter’ is an example how devices can control the flow of light.
For the understanding of the functioning and design of such devices, a major topic is to construct suitable conditions at an artificial boundary that do not introduce additional reflections that may disturb the light near the device, but, on the other hand, will confine the domain of interest to a finite domain around the device. Introducing such boundary conditions, new mathematical problems appear; the solutions are relevant for theoretical insight, but just as well as for the design of numerical simulation tools.
For another part, research in optics deals with propagational aspects in nonlinear fiber optics, when light pulses are send over large distances (across oceans). This reserach closely resembles research on surface water waves.
An intense collaboration is established with the Light Wave Device group at Applied Physics UT, and with optical industry Kymata/Arcatel. The research is part of the UT-spearhead institute MESA+.
Research in fluid dynamics
concerns the study and simulation of non-linear deformations
of surface water waves.
For simulational purposes a ‘numerical wave tank’ has been designed based on the full irrotational surface wave equations. Asymptotic models, like NLS (NonLinear Schrodinger) - and KdV (Korteweg and De Vries) -type of equations, are used for theoretical investigations. The results form also the basis of an Analytic Wave Code with which wave fields can be predicted at all places from measurements at a single or few places.
Also the interaction between currents and surface waves is being investigated.
This research is executed in collaboration with MARIN (Maritime Research Institute Netherlands) and Pusat Matematika P4M at Institut Teknologi Bandung, Indonesia, and is part of activities of the UT-Research Institute TIM.
Much of the research in these two different areas is comparable from a mathematical point of view. The reason is that the basic equations for (inviscid) fluid dynamics and electromagnetism, describing water waves as well as and optical waves, have the same mathematical structure: a dynamical system with a Poisson structure.