On Singularities of Certain Non-linear Second-Order Ordinary Differential Equations

Abstract

The method of blowing up points of indeterminacy of certain systems of two ordinary differential equations is applied to obtain information about the singularity structure of the solutions of the corresponding non-linear differential equations. We first deal with the so-called Painlevé example, which passes the Painlevé test, but the solutions have more complicated singularities. Resolving base points in the equivalent system of equations we can explain the complicated structure of singularities of the original equation. The Smith example has a solution with non-isolated singularity, which is an accumulation point of algebraic singularities. Smith’s equation can be written as a system in two ways. We show that the sequence of blow-ups for both systems can be infinite. Another example that we consider is the Painlevé-Ince equation. When the usual Painlevé analysis is applied, it possesses both positive and negative resonances. We show that for three equivalent systems there is an infinite sequence of blow-ups and another one that terminates, which further gives a Laurent expansion of the solution around a movable pole. Moreover, for one system it is even possible to obtain the general solution after a sequence of blow-ups.

Regularising Transformations for Complex Differential Equations with Movable Algebraic Singularities

“…An example where the sequence of blow-ups does not terminate is given by Smith's equation y + 4y 3 y + y = 0, which is not of Hamiltonian form. This was noted by the authors in [6] and is a hint that for this equation more complicated movable singularities exist than considered in this article. In fact, as noted earlier, Smith himself showed that there do exist singularities besides the algebraic poles (18), which are are known to be accumulation points of such algebraic poles.…”

Regularising transformations for the $$(n,\,n+1)$$-Liénard equations

“…The cascade from p 5 leads to more and more complicated expressions for points of indeterminacy of the corresponding systems and it does not stop after a reasonable number of steps. In general, as discussed in [5], the infinite cascades possibly indicate the presence of more complicated singularities or some special solutions. Blowing up p 2 we find the point p 3 = (u 2 = 0, v 2 = 2/ f 1 ) on the exceptional line.…”

“…As already mentioned in [5], Liénard type equations may possess cascades which do not finish after a reasonable number of steps. In some cases expressions of consequitive points become very cumbersome and the complexity of symbolic computations rise singificantly.…”

“…In other cases the form of the equations does not change much and similar points appear over and over again. One such example is the Smith equation y = 4y 3 y + y discussed in [3,5]. Another example is the maximum balance equation y = μy n y + ν y 2n+1 from [3] with n = 2: y = μy 2 y + ν y 5 .…”

“…The geometric approach for the Painlevé equations was developed in the papers [9,10] (see also [7]). The method of blowing up points of indeterminacy of certain systems of two ordinary differential equations was also applied to obtain information about the singularity structure of solutions of the corresponding non-linear differential equations in [5,8]. Originally, the notion of a blow-up comes from algebraic geometry.…”