On the connection formulas of Painlevé transcendents

Abstract

The six Painleve equations were first discovered around the beginning of the twentieth century by Painleve, Gambier and their colleagues in an investigation of nonlinear second-order ordinary differential equations. They are solutions to certain nonlinear second-order ordinary differential equations in the complex plane with the Painleve property (the only movable singularities are poles). They are not generally solvable in terms of elementary functions. Although first discovered from strictly mathematical considerations, the six Painleve equations have arisen in a variety of important physical applications including statistical mechanics, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics and fibre optics. It is now becoming clear that the Painleve transcendents play the same role in nonlinear mathematical physics that the clsssical special functions, such as Airy functions, Bessel functions, etc., play in linear physics. In this thesis, we use the method of "uniform asymptotics" proposed by Bassom, Clarkson, Law and McLeod (Arch. Rat. Mech. Anal., 1998) to consider the connection problem for the sine-Gordon PIII equation

which is the most commonly studied case among all general third Painleve transcendents. Using the same method, we also provide a simpler and more rigorous proof of the connection formulas of the Clarkson-McLeod solution for the fourth Painleve transcendent

which was established by Its and Kapaev (J. Phys. A: Math. Gen., 1998) via the isomonodromy and Riemann Hilbert methods.