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.

In a recent paper, X. F. Yang proved a uniqueness theorem on inverse nodal problems that links to inverse spectral theory, on one hand, and reduces the redundancy of the classical inverse nodal problems, on the other hand. In this note we improve Yang's theorem by weakening its conditions and simplifying its proof. We also discuss variants of Yang's theorem. ᮊ

The inverse nodal problem on the Sturm-Liouville operator is the problem of finding the potential function q and boundary conditions α,β using the nodal sequence {xk(n)}. In this paper, we show that the space of all (q,α,β) such that ∫01q = 0, under a certain metric, is homeomorphic to the partition set of all asymptotically equivalent nodal sequences induced by an equivalence relation. As a consequence, the inverse nodal problem, when defined on the partition set of admissible sequences induced by the same equivalence relation, is well posed. Let Φ be the homeomorphism, which we call a nodal map. We find that Φ is still a homeomorphism when the corresponding metrics are magnified by the derivatives of q, whenever q is CN. Our method depends heavily on the explicit asymptotic expressions of the nodal points and nodal lengths.

We study the issues of the reconstruction and stability of the inverse nodal problem for the one-dimensional p-Laplacian eigenvalue problem. A key step is the application of a modified Prüfer substitution to derive a detailed asymptotic expansion for the eigenvalues and nodal lengths. Two associated Ambarzumyan problems are also solved.