Applications of Nevanlinna theory to differential and difference equations

Strong hints as to whether certain types of equations are integrable (in some sense solvable) can be obtained by looking at the behaviour of solutions in the complex domain, even when we are only concerned with finding real solutions. For differential equations the most widely used property of this type is the Painlevé property. An ordinary differential equation has the Painlevé property if all solutions are single-valued about all movable singularities. Although proving that a given equation actually has the Painlevé property is sometimes difficult, there are very powerful but simple methods that can be applied to very general equations that will show that they do not have the Painlevé property. These classifications are based on purely local methods such as series expansions. Unfortunately the Painlevé property is destroyed by almost any rational change of dependent variable. However, although the transformed equation will have movable branch points, the global branching structure will remain simple. In particular, if the solution has no fixed singularities then it is algebroid (algebriac over the meromorphic functions) so its Riemann surface has only a finite number of sheets over any point. The first part of this project is to develop methods to determine when an equation possesses algebroid solutions. This should be a more sensitive test than the usual Painlevé analysis. Local series expansions will not be enough here and more global methods are required. Here Nevanlinna theory, the theory of the value distribution of meromorphic functions, will play a central role.

The second part of the project again uses Nevanlinna theory as a way of detecting integrable equations, but this time we will consider differential-delay equations, e.g., equations of the form F(z,y(z+1),y(z-1),y(z),y'(z))=0. Based on earlier work on difference equations, differential-delay equations that admit meromorphic solutions that are finite-order (i.e., not too complicated) in the sense of Nevanlinna theory of a particular type will be classified. Very little work on the integrability of differential-delay equations has been carried out. Differential-delay equations appear in many models, especially in mathematical biology.

Looking beyond the immediate beneficiaries in the fields of integrable systems and complex analysis, this project could potentially impact any scientist or other quantitative researcher (e.g. in economics or finance) looking to identify "more solvable" models of various phenomena. Within mathematics, geometry is a natural area in which many integrable systems have been (re-)discovered and the integrability of the differential equations involved have led to greater understanding. The algebroid methods of the current project should give a more sensitive detector for such equations.

Differential-delay equations appear in many applications such as mathematical biology and mathematical finance - areas in which some kind of a seasonal effects play significant roles. Although one would not expect that many of the differential-delay equations that arise in such applications are integrable, the study of integrable differential-delay equations should lead to significant advances in the general theory. This is certainly what has happened historically in the case of ordinary and partial differential equations, discrete equations and differential-discrete equations. Often the detailed study of particularly nice examples leads to a better understanding and subsequent development of the general theory. Very little is known about nonlinear differential-delay equations and the identification of the nicest (i.e. integrable) genuinely nonlinear example could be a significant step towards a more complete general theory of these equations, which would have impact on such applications as already mentioned. Presumably the timeframe involved here would be on the order of decades.

In the case of differential equations, many important systems can be viewed as perturbations of intergable equations and this has led to many advances in physics and engineering. The possibility of viewing certain differential-delay equations as perturbed integrable equations could lead to major advances.

Other potential impact would be via symbolic programmers. Several people, such as Willy Hereman at the Colorado School of Mines, have developed symbolic computational packages based on the Painlevé property. The algebroid and differential-delay work of the current project could lead to test that can be implemented on a computer and made available to a wider group of users. The timescale here could be potentially five years.