Visualizing Spacetime Curvature via Gradient Flows II: An Example of the Construction of a Newtonian analogue

This is the first in a series of papers in which the gradient flows of fundamental curvature invariants are used to formulate a visualization of curvature. We start with the construction of strict Newtonian analogues (not limits) of solutions to Einstein's equations based on the topology of the associated gradient flows. We do not start with any easy case. Rather, we start with the Curzon - Chazy solution, which, as history shows, is one of the most difficult exact solutions to Einstein's equations to interpret physically. We show that the entire field of the Curzon - Chazy solution, up to a region very "close" to the the intrinsic singularity, strictly represents that of a Newtonian ring, as has long been suspected. In this regard, we consider our approach very successful. As regrades the local structure of the singularity of the Curzon - Chazy solution within a fully general relativistic analysis, however, whereas we make some advances, the full structure of this singularity remains incompletely resolved.Comment: 12 pages twocolumn revtex 4-1 9 figures. Expanded and correcte

Movable algebraic singularities of second-order ordinary differential equations

Any nonlinear equation of the form y''=\sum_{n=0}^N a_n(z)y^n has a (generally branched) solution with leading order behaviour proportional to (z-z_0)^{-2/(N-1)} about a point z_0, where the coefficients a_n are analytic at z_0 and a_N(z_0)\ne 0. We consider the subclass of equations for which each possible leading order term of this form corresponds to a one-parameter family of solutions represented near z_0 by a Laurent series in fractional powers of z-z_0. For this class of equations we show that the only movable singularities that can be reached by analytic continuation along finite-length curves are of the algebraic type just described. This work generalizes previous results of S. Shimomura. The only other possible kind of movable singularity that might occur is an accumulation point of algebraic singularities that can be reached by analytic continuation along infinitely long paths ending at a finite point in the complex plane. This behaviour cannot occur for constant coefficient equations in the class considered. However, an example of R. A. Smith shows that such singularities do occur in solutions of a simple autonomous second-order differential equation outside the class we consider here

Discrete and Continuous Linearizable Equations

We study the projective systems in both continuous and discrete settings. These systems are linearizable by construction and thus, obviously, integrable. We show that in the continuous case it is possible to eliminate all variables but one and reduce the system to a single differential equation. This equation is of the form of those singled-out by Painlev\'e in his quest for integrable forms. In the discrete case, we extend previous results of ours showing that, again by elimination of variables, the general projective system can be written as a mapping for a single variable. We show that this mapping is a member of the family of multilinear systems (which is not integrable in general). The continuous limit of multilinear mappings is also discussed.Comment: Plain Tex file, 14 pages, no figur

A new method to test discrete Painlev\'e equations

Necessary discretization rules to preserve the Painlev\'e property are stated. A new method is added to the discrete Painlev\'e test, which perturbs the continuous limit and generates infinitely many no-log conditions.Comment: 12 pages, no figure, standard Latex, to appear in Physics Letters

Gap Probabilities for Edge Intervals in Finite Gaussian and Jacobi Unitary Matrix Ensembles

The probabilities for gaps in the eigenvalue spectrum of the finite dimension $N \times N$ random matrix Hermite and Jacobi unitary ensembles on some single and disconnected double intervals are found. These are cases where a reflection symmetry exists and the probability factors into two other related probabilities, defined on single intervals. Our investigation uses the system of partial differential equations arising from the Fredholm determinant expression for the gap probability and the differential-recurrence equations satisfied by Hermite and Jacobi orthogonal polynomials. In our study we find second and third order nonlinear ordinary differential equations defining the probabilities in the general $N$ case. For N=1 and N=2 the probabilities and thus the solution of the equations are given explicitly. An asymptotic expansion for large gap size is obtained from the equation in the Hermite case, and also studied is the scaling at the edge of the Hermite spectrum as $N \to \infty$, and the Jacobi to Hermite limit; these last two studies make correspondence to other cases reported here or known previously. Moreover, the differential equation arising in the Hermite ensemble is solved in terms of an explicit rational function of a {Painlev\'e-V} transcendent and its derivative, and an analogous solution is provided in the two Jacobi cases but this time involving a {Painlev\'e-VI} transcendent.Comment: 32 pages, Latex2

Constructing Integrable Third Order Systems:The Gambier Approach

We present a systematic construction of integrable third order systems based on the coupling of an integrable second order equation and a Riccati equation. This approach is the extension of the Gambier method that led to the equation that bears his name. Our study is carried through for both continuous and discrete systems. In both cases the investigation is based on the study of the singularities of the system (the Painlev\'e method for ODE's and the singularity confinement method for mappings).Comment: 14 pages, TEX FIL

On the discrete and continuous Miura Chain associated with the Sixth Painlevé Equation

A Miura chain is a (closed) sequence of differential (or difference) equations that are related by Miura or B\"acklund transformations. We describe such a chain for the sixth Painlev\'e equation (\pvi), containing, apart from \pvi itself, a Schwarzian version as well as a second-order second-degree ordinary differential equation (ODE). As a byproduct we derive an auto-B\"acklund transformation, relating two copies of \pvi with different parameters. We also establish the analogous ordinary difference equations in the discrete counterpart of the chain. Such difference equations govern iterations of solutions of \pvi under B\"acklund transformations. Both discrete and continuous equations constitute a larger system which include partial difference equations, differential-difference equations and partial differential equations, all associated with the lattice Korteweg-de Vries equation subject to similarity constraints

Non-Schlesinger Deformations of Ordinary Differential Equations with Rational Coefficients

We consider deformations of $2\times2$ and $3\times3$ matrix linear ODEs with rational coefficients with respect to singular points of Fuchsian type which don't satisfy the well-known system of Schlesinger equations (or its natural generalization). Some general statements concerning reducibility of such deformations for $2\times2$ ODEs are proved. An explicit example of the general non-Schlesinger deformation of $2\times2$-matrix ODE of the Fuchsian type with 4 singular points is constructed and application of such deformations to the construction of special solutions of the corresponding Schlesinger systems is discussed. Some examples of isomonodromy and non-isomonodromy deformations of $3\times3$ matrix ODEs are considered. The latter arise as the compatibility conditions with linear ODEs with non-singlevalued coefficients.Comment: 15 pages, to appear in J. Phys.