NEW EXACT-SOLUTIONS OF THE DISCRETE 4TH PAINLEVE EQUATION

In this paper we derive a number of exact solutions of the discrete eq uation X(n+1)X(n-1) + X(n)(X(n+1) + X(n-1)) = [-2z(n)X(n)3 + (eta - 3d elta-2-z(n)2)x(n)2 + mu2]/(x(n) + z(n) + gamma)(X(n) + z(n) - gamma), where z(n) = ndelta and eta, delta, mu and gamma are constants. In an appropriate limit this equation reduces to the fourth Painleve (PIV) e quation d2w/dz2 = (1/2w) (dw/dz)2 + 3/2 w3 + 4zw2 + 2(z2 - alpha)w + b eta/w, where alpha and beta are constants, and it is commonly referred to as the discretised fourth Painleve equation. A suitable factorisat ion of this equation facilitates the identification of a number of sol utions which take the form of ratios of two polynomials in the variabl e z(n). Limits of these solutions yield rational solutions of this PIV equation. It is also known that there exist exact solutions of this P IV equation that are expressible in terms of the complementary error f unction and in this article we show that a discrete analogue of this f unction can be obtained by analysis of the discrete equation above.