Application of Uniform Asymptotics to the Second Painlevé Transcendent

Abstract

In this work we propose a new method for investigating connection problems for the class of nonlinear second-order differential equations known as the Painlevé equations. Such problems can be characterized by the question as to how the asymptotic behaviours of solutions are related as the independent variable is allowed to pass towards infinity along 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 WKB solutions. However, the implementation of these methods often tends to be heuristic in nature and so the task of rigorising the process is complicated. The method we propose here develops uniform approximations to solutions. This removes the need to match solutions, is rigorous, and can lead to the solution of connection problems with minimal computational effort.Our method is reliant on finding uniform approximations of differential equations of the generic formas the complex-valued parameter ξ → ∞. The details of the treatment rely heavily on the locations of the zeros of the function F in this limit. If they are isolated then a uniform approximation to solutions can be derived in terms of Airy functions of suitable argument. On the other hand, if two of the zeros of F coalesce as |ξ| → ∞ then an approximation can be derived in terms of parabolic cylinder functions. In this paper we discuss both cases, but illustrate our technique in action by applying the parabolic cylinder case to the "classical" connection problem associated with the second Painlevé transcendent. Future papers will show how the technique can be applied with very little change to the other Painlevé equations, and to the wider problem of the asymptotic behaviour of the general solution to any of these equations.

Application of uniform asymptotics to the connection formulas of the fifth Painlevé equation

“…For large x, equation (2.9) has two turning points In line with the idea of uniform asymptotics in [4], we define a number α such that…”

“…Along the same lines we may find the work of Olver [20] and Dunster [6] for coalescing turning points. Initially in [4], PII has been taken as an example to illustrate the method. While the difficulty in extending the techniques for PII to other transcendents is also acknowledged by the authors of [4, p.244].…”

Application of connection formula on eliminating plausible asymptotic representation of the painlevé transcendents

“…In next section, we develop a plausible asymptotic representation of (PIV) as ! e 5i=4 1: In Section 3, we use the uniform asymptotics 854 Y. LU method developed by Bassom et al [5] to find the monodromy data corresponding to this asymptotics and the one in Theorem 1. In last section, we find the connection formulae of these two asymptotics by using the monodromy data found in Section 3, and then find a contradiction.…”

“…entries a, b, c, and d are independent of : (3) All of the monodromy data of the system can be expressed in terms two of the four entries a, b, c and d.Now we start to use the uniform asymptotics method developed by Bassom et al[5] to find the entries a and b corresponding to a solution of (PIV) of the form and (3.4), we can find an asymptotic expression for v and u:…”

Discrete Integrable Systems Associated With the Unitary Matrix Model

“…In this paper, we show that the orthogonal polynomials on the unit circle associated with the unitary matrix model satisfy the linear problems for the discrete PainlevéII hierarchy, alternate discrete PainlevéII and discrete MKdV hierarchy. In a later paper, we will use isomonodromy deformation and uniform asymptotics (as developed in [2]) to discuss the linear problems and the solution to the discrete equations.…”