Differential equations play a key role in describing a wide range of dynamical processes, from mechanical systems, fluid dynamics, optical phenomena and quantum mechanics amongst many others, where they relate the state of a physical system to the rate of change that the system is momentarily undergoing.
It is important in applications to know when a system has solutions that are, in a certain mathematical sense, well-behaved, or whether the solutions exhibit chaotic behaviour. Although in applications one is usually interested in the time-evolution of a system, time being a real, 1-dimensional parameter, it has long been understood that the nature of the solutions of an equation is best explained when time is considered as member of an extended number system, mathematically realised as points in a 2-dimenional plane, known as the complex numbers.
The solutions of a differential equation, considered in this 2-dimensional complex plane, can have singularities, i.e. points where a solution becomes infinite or is otherwise ill-defined. The nature of these singularities, in turn, has an impact on the behaviour of the solutions in the 1-dimensional time parameter space. It allows one to determine whether an equation is integrable, i.e. exactly solvable in a certain mathematical sense.
This project is about investigating the nature of the singularities of certain wide classes of equations that are motivated from physical applications. To investigate the singularities, we will consider the solutions of the equations in spaces of the dependent variables (those variables describing the state of a system and its rate of change) to which certain points have been added to include points where these become infinite. In this way, one can investigate what happens when the solutions develop a singularity. It turns out, however, that even in these augmented spaces there are points at which the rate of change of a system is indeterminate, which happens when both the numerator and the denominator of the fractions expressing the rate of change approach zero at the same time. Such 'base points' can be removed by a well-defined mathematical procedure known as a blow-up. The blow-up of a point is a geometrical notion which adds further points to the solution space by introducing an 'exceptional line', each point of which corresponds to a direction emerging from the point in question. In this way, the solutions of an equation are separated out over the exceptional line, making it possible to investigate them further. The 'exceptional line' introduced by the blow-up, when considered as an object in complex geometry (where the coordinates take complex numbers as values) turns out to be equivalent to a sphere. In this way, the original point at which the rate of change of the solution was ill-defined, becomes inflated to a sphere, hence the name blow-up for this process.
It turns out that for most equations, a single blow-up of the equation at one point is not sufficient to render the equation free of indeterminacies, i.e., even after one blow-up has been performed, further ones may be necessary. For the Painlevé equations, an important class of equations in mathematical physics, it turns out that a total of 9 blow-ups is required to bring these equations into a form where no points with indeterminate behaviour remain. The solution space thus constructed, first obtained by the Japanese mathematician K. Okamoto, is called the 'space of initial values' of the equation. For the Painlevé equations, using this space one can explain the nature of the singularities, which in this case are poles: points at which the solutions tend to infinity in a controlled way.
In the proposed project, we will apply the method of constructing the space of initial values to much wider classes of equations to investigate how this method can be utilised to determine the nature of more complicated singularities that certain differential equations can exhibit.
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