APPLICATION OF UNIFORM ASYMPTOTICS TO THE 2ND PAINLEVE TRANSCENDENT

In this work we propose a new method for investigating connection prob lems for the class of nonlinear second-order differential equations kn own as the Painleve equations. Such problems can be characterized by t he question as to how the asymptotic behaviours of solutions are relat ed as the independent variable is allowed to pass towards infinity alo ng different directions in the complex plane. Connection problems have been previously tackled by a variety of methods. Frequently these are based on the ideas of isomonodromic deformation and the matching of W KB solutions. However, the implementation of these methods often tends to be heuristic in nature and so the task of rigorising the process i s complicated. The method we propose here develops uniform approximati ons to solutions. This removes the need to match solutions, is rigorou s, and can lead to the solution of connection problems with minimal co mputational effort. Our method relies on finding uniform approximation s of differential equations of the generic form [GRAPHICS] as the comp lex-valued parameter xi --> infinity. The details of the treatment rel y heavily on the locations of the zeros of the function F in this limi t. IF they are isolated, then a uniform approximation to solutions can be derived in terns of Airy functions of suitable argument. On the ot her hand, if two of the zeros of F coalesce as \xi\ --> infinity, then an approximation can be derived in terms of parabolic cylinder functi ons. In this paper we discuss both eases, but illustrate our technique in action by applying the parabolic cylinder case to the ''classical' ' connection problem associated with the second Painleve transcendent. Future papers will show how the technique can be applied with very li ttle change to the other Painleve equations, and to the wider problem of the asymptotic behaviour of the general solution to any of these eq uations.