We introduce a rigorous methodology for studying the Riemann-Hilbert problems associated
with certain integrable nonlinear ordinary differential equations. For concreteness we …

The power spectrum is a Fourier series statistic associated with the covariances of the
displacement from average positions of the members of an eigen-sequence. When this …

The characteristic feature of the so‐called Painlevé test for integrability of an ordinary (or
partial) analytic differential equation, as usually carried out, is to determine whether all its …

The short-distance expansion of the tau function of the radial sine-Gordon/Painlevé III
equation is given by a convergent series which involves irregular conformal blocks and …

The application of local Lie point transformation groups to the solution of partial differential
equations is reviewed. The method of symmetry reduction is presented as an algorithm …

The Bonnet theorem characterizes surfaces via the coefficients eu, H, Q of their fundamental
forms. These coefficients are not independent and are subject to the Gauss-Codazzi …

In this article our concern is with the third Painlevé equation d2y/dx2=(1/y)(dy/dx)
2−(1/x)(dy/dx)+(αy2+ β)/x+ γy3+ δ/y where α, β, γ, and δ are arbitrary constants. It is well …

The main functions covered in this chapter are cuspoid catastrophes ΦK (t; x); umbilic
catastrophes with codimension three Φ (E)(s, t; x), Φ (H)(s, t; x); canonical integrals ΨK (x), Ψ …

We evaluate explicitly, in terms of the Cauchy data, the constant pre-factor in the large
asymptotics of the Painlevé III tau-function. Our result proves the conjectural formula for this …

In this work we collect together our connection results for asymptotics as t→+ 0 and t→+∞ of
general and some special solutions of the fifth Painlevé equation. Kee** in mind already …