In this paper we define the Painleve property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Backlund transforms, the linearizing transforms, and the Lax pairs of three well‐known partial differential equations (Burgers’ equation, KdV equation, and the modified KdV equation). This indicates that the Painleve property may provide a unified description of integrable behavior in dynamical systems (ordinary and partial differential equations), while, at the same time, providing an efficient method for determining the integrability of particular systems.

The Painlevé property for partial differential equations

http://staff.ustc.edu.cn/~wzhao7/c_index_files/main.files/lss.pdf

Large scale structure of the universe and cosmological perturbation theory

Random-matrix theories in quantum physics : common concepts

R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

We review the development of random-matrix theory (RMT) during the last decade. We emphasize both the theoretical aspects, and the application of the theory to a number of fields. These comprise chaotic and disordered systems, the localization problem, many-body quantum systems, the Calogero-Sutherland model, chiral symmetry breaking in QCD, and quantum gravity in two dimensions. The review is preceded by a brief historical survey of the developments of RMT and of localization theory since their inception. We emphasize the concepts common to the above-mentioned fields as well as the great diversity of RMT. In view of the universality of RMT, we suggest that the current development signals the emergence of a new "statistical mechanics": Stochasticity and general symmetry requirements lead to universal laws not based on dynamical principles.

Random Matrix Theories in Quantum Physics: Common Concepts

We develop here two aspects of the connection between nonlinear partial differential equations solvable by inverse scattering transforms and nonlinear ordinary differential equations (ODE) of P‐type (i.e., no movable critical points). The first is a proof that no solution of an ODE, obtained by solving a linear integral equation of a certain kind, can have any movable critical points. The second is an algorithm to test whether a given ODE satisfies necessary conditions to be of P‐type. Often, the algorithm can be used to test whether or not a given nonlinear evolution equation may be completely integrable.

A connection between nonlinear evolution equations and ordinary differential equations of P‐type. II

The Painlevé property and singularity analysis of integrable and non-integrable systems

We present an integrability criterion for discrete-time systems that is the equivalent of the Painlev\'e property for systems of a continuous variable. It is based on the observation that for integrable mappings the singularities that may appear are confined, i.e., they do not propagate indefinitely when one iterates the mapping. Using this novel criterion we show that there exists a family of nonautonomous integrable mappings which includes the discrete Painlev\'e equations by ${\mathit{P}}_{\mathrm{I}}$, recently derived in a model of two-dimensional quantum gravity, and ${\mathit{P}}_{\mathrm{II}}$, obtained as a similarity reduction of the integrable modified Korteweg--de Vries lattice. These systems possess Lax pairs, a well-known integrability feature.

/pdf/do-integrable-mappings-have-the-painleve-property-vw125aee1r.pdf

Do integrable mappings have the Painlevé property

Nonlinear evolution equations and ordinary differential equations of painlevè type

We present discrete forms of the Painlev\'e transcendental equations ${\mathit{P}}_{\mathrm{III}}$,${\mathit{P}}_{\mathrm{IV}}$, and ${\mathit{P}}_{\mathrm{V}}$ that complement the list of the already known ${\mathit{P}}_{\mathrm{I}}$ and ${\mathit{P}}_{\mathrm{II}}$. These, most likely integrable, nonautonomous mappings go over to the usual Painlev\'e equations in the continuous limit, while in the autonomous limit we recover discrete system that belong to the integrable family of Quispel et al. Finally, we show that the discrete Painlev\'e mappings satisfy the same reduction relations as the continuous Painlev\'e transcendents, namely, ${\mathit{P}}_{\mathrm{V}}$\ensuremath{\rightarrow}{${\mathit{P}}_{\mathrm{III}}$, ${\mathit{P}}_{\mathrm{IV}}$}\ensuremath{\rightarrow}${\mathit{P}}_{\mathrm{II}}$\ensuremath{\rightarrow}${\mathit{P}}_{\mathrm{I}}$.

Alfred Ramani

Papers

A connection between nonlinear evolution equations and ordinary differential equations of P‐type. II

The Painlevé property and singularity analysis of integrable and non-integrable systems

Do integrable mappings have the Painlevé property

Nonlinear evolution equations and ordinary differential equations of painlevè type

Discrete versions of the Painlevé equations.