A rigorous proof of an exponentially small estimate for a boundary value arising from an ordinary differential equation

Abstract

SynopsisThe equationhas a solution y(t) which is non-oscillating on the interval (0, ∞) and has the asymptotic expansionEach term of this expansion is even in t so that formally is zero to all orders of ɛ. The estimate of has been obtained by Byatt-Smith [3] who corrects (2) in the complex plane near t = i where the series ceases to be valid. This requires asolution of the equationthe equation for the first Painlevé transcedent. Here we prove rigorously that this method gives the correct asymptotic estimatewhereThe proof involves converting (1) and (3) to integral equations. The existence and uniqueness of these integral equations are established by use of the contraction mapping theorem. We also prove that the appropriate solution to (3) provides a uniformly valid approximation to (2) over a suitably defined region of the complex plane.We also consider the connection problem for the oscillatory solutions of (1) which have asymptotic expansionswhere Ã+, Ã− φ+, and φ− are constants. The connection problem is to determine the asymptotic expansion at +∞ of a solution which has a given asymptotic expansion at −∞. In other words, we wish to find (Ã+,φ+) as a function of Ã−and φ−. We prove that there is a unique solution to the connection problem, provided Ã− is small enough, and obtain bounds on the estimate of Ã+