Backlund Transformations and Hierarchies of Exact Solutions for the Fourth Painleve Equation and their Application to Discrete Equations

Abstract

In this paper we describe Bäcklund transformations and hierarchies of exact solutions for the fourth Painlevé equation (PIV) $${\d^2 w\over\d z^2}={1\over2w}\left(\d w\over\d z\right)^2 + {3\over2}w^3 + 4zw^2 + 2(z^2-α)w+{β\over w},\eqno(1){\hbox to 16pt{\hfill}}$$ with $α$, $β$ constants. Specifically, a nonlinear superposition principle for PIV, hierarchies of solutions expressible in terms of complementary error or parabolic cylinder functions as well as rational solutions will be derived. Included amongst these hierarchies are solutions of (1) for which $α=\pm\tfr12n$ and $β=-\tfr12n^2$, with $n$ an integer. These particular forms arise in quantum gravity and also satisfy a discrete analogue of the first Painlevé equation. We also obtain a number of exact solutions of the discrete fourth Painlevé equation $$x_{n+1}x_{n-1}+x_n(x_{n+1}+x_{n-1})= {-2z_nx_n^3+(η-3δ^{-2}-z_n^2)x_n^2+μ^2\over (x_n+z_n+γ)(x_n+z_n-γ)},\eqno(2){\hbox to 16pt{\hfill}}$$}% {\narrower\noindent\baselineskip=12pt where $z_n=nδ$ and $η$, $δ$, $μ$ and $γ$ are constants, which, in an appropriate limit, reduces to PIV (1). A suitable factorisation of (2) facilitates the identification of a number of solutions which take the form of ratios of two polynomials in the variable $z_n$ and the limits of these solutions yield rational solutions of (1).