BACKLUND-TRANSFORMATIONS AND SOLUTION HIERARCHIES FOR THE 4TH PAINLEVE EQUATION

In this paper our concern is with solutions w (z; alpha, beta) of the fourth Painleve equation (PIV), where alpha and beta are arbitrary rea l parameters. it is known that PIV admits a variety of solution types and here we classify and characterise these. Using Backlund transforma tions we describe a novel method for efficiently generating new soluti ons of PIV from known ones. Almost all the established Backlund transf ormations involve differentiation of solutions and since all but a ver y few solutions of PIV are given by extremely complicated formulae, th ose transformations which require differentiation in this way are very awkward to implement in practice. Depending on the values of the para meters alpha and beta, PIV can admit solutions which may either be exp ressed as the ratio of two polynomials in z, or can be related to the complementary error or parabolic cylinder functions; in fact, all exac t solutions of PIV are thought to fall in one of these three hierarchi es. We show how, given a few initial solutions, it is possible to use the structures of the hierarchies to obtain many other solutions. In o ur approach we derive a nonlinear superposition formula which relates three solutions of PIV; the principal attraction is that the process i nvolves only algebraic manipulations so that, in particular, no differ entiation is required. We investigate the properties of our computed s olutions and illustrate that they have a large number of physical appl ications.