Bäcklund Transformations and Solution Hierarchies for the Fourth Painlevé Equation

Abstract

In this paper our concern is with solutions w ( z ; α , β ) of the fourth Painlevé equation (PIV), where α and β are arbitrary real parameters. It is known that PIV admits a variety of solution types and here we classify and characterise these. Using Backlund transformations we describe a novel method for efficiently generating new solutions of PIV from known ones. Almost all the established Bäcklund transformations involve differentiation of solutions and since all but a very few solutions of PIV are given by extremely complicated formulae, those transformations which require differentiation in this way are very awkward to implement in practice. Depending on the values of the parameters α and β , PIV can admit solutions which may either be expressed as the ratio of two polynomials in z , or can be related to the complementary error or parabolic cylinder functions; in fact, all exact solutions of PIV are thought to fall in one of these three hierarchies. We show how, given a few initial solutions, it is possible to use the structures of the hierarchies to obtain many other solutions. In our approach we derive a nonlinear superposition formula which relates three solutions of PIV; the principal attraction is that the process involves only algebraic manipulations so that, in particular, no differentiation is required. We investigate the properties of our computed solutions and illustrate that they have a large number of physical applications.