Backlund transformations and solution hierarchies for the third Painleve equation

In this article our concern is with the third Painleve equation (1) d(2)y/dx(2) = 1/y (dy/dx)(2) - 1/x dy/dx + alpha y + beta/x + gamma y(3) + delta/y, where alpha, beta, gamma, and delta are arbitrary constants. It is well known that this equation admits a variety of types of solution and here we classify and characterize many of these. Depending on the values of the parameters the third Painleve equation can admit solutions that may be either expressed as the ratio of two polynomials in either x or x(1/3) or related to certain Bessel functions. It is thought that all exact solutions of (1) can be categorized into one or other of these hierarchies. We show how, given a few initial solutions, it is possible to use the underlying structures of these hierarchies to obtain many other solutions. In addition, we show how this knowledge concerning the continuous third Painleve equation (1) can be adapted and used to derive exact solutions of a suitable discretized counterpart of (1). Both the continuous and discrete solutions we find are of potential importance as it is known that the third Painleve equation has a large number of physically significant applications.