The Painlevé-Kowalevski and Poly-Painlevé Tests for Integrability

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 solutions are single-valued by local analysis near individual singular points of solutions. This test, interpreted flexibly, has been quite successful in spite of various evident flaws. We review the Painleve test in detail and then propose a more robust and generally more appropriate definition of integrability: a multivalued function is accepted as an integral if its possible values (at any given point in phase space) are not dense. This definition is illustrated and justified by examples, and a widely applicable method (the poly-Painlevé method) of testing for it is presented, based on asymptotic analysis covering several singularities simultaneously.     

Rational Solutions of Painlevé Equations

With Bäcklund transformations, we construct explicit solutions of Painlevé equations 2 and 4. Independently, we find solutions of degenerate cases of equations 3 and 5. The six Painlevé transcendents are referred to as 1–6.     

On One-Parameter Families of Painlevé III

Albrecht, Mansfield, and Milne developed a direct method with which one can calculate special integrals of polynomial type (also known as one-parameter family conditions, Darboux polynomials, eigenpolynomials, or algebraic invariant curves) for nonlinear ordinary differential equations of polynomial type. We apply this method to the third Painlevé equation and prove that for the generic case, the set of known one-parameter family conditions is complete.     

Painlevé VI and the Unitary Jacobi Ensembles

The six Painlevé transcendants which originally appeared in the studies of ordinary differential equations have been found numerous applications in physical problems. The well‐known examples among which include symmetry reduction of the Ernst equation which arises from stationary axial symmetric Einstein manifold and the spin‐spin correlation functions of the two‐dimensional Ising model in the work of McCoy, Tracy, and Wu. The problem we study in this paper originates from random matrix theory, namely, the smallest eigenvalues distribution of the finite n Jacobi unitary ensembles which was first investigated by Tracy and Widom. This is equivalent to the computation of the probability that the spectrum is free of eigenvalues on the interval . Such ensembles also appears in multivariate statistics known as the double‐Wishart distribution. We consider a more general model where the Jacobi weight is perturbed by a discontinuous factor and study the associated finite Hankel determinant. It is shown that the logarithmic derivative of Hankel determinant satisfies a particular σ‐form of Painlevé VI, which holds for the gap probability as well. We also compute exactly the leading term of the gap probability as .     

A Property of the Ansatz of Hirota's Method for Quasilinear Parabolic Equations

By using the recently discovered new invariant properties of the ansatz of R. Hirota's method, we prove that the classes of linear fractional solutions to some nonlinear equations are closed. This allows us to construct new solutions for a chosen class of dissipative equations. This algorithm is similar to the method of dressing the solutions of integrable equations. The equations thus obtained imply a compatibility condition and are known as a nonlinear Lax pair with variable coefficients. So we propose a method for constructing such pairs. To construct solutions of a more complicated form, we propose to use the property of zero denominators and factorized brackets, which has been discovered experimentally. The expressions thus constructed are said to be quasi-invariant. They allow us to find true relations between the functions contained in the ansatz, to correct the ansatz, and to construct a solution. We present some examples of new solutions constructed following this approach. Such solutions can be used for majorizing in comparison theorems and for modeling phase processes and process in neurocomputers. A program for computing solutions by methods of computer algebra is written. These techniques supplement the classical methods for constructing solutions by using their group properties.     

Bäcklund transformations for higher order Painlevé equations

We present a new generalized algorithm which allows the construction of Bäcklund transformations (BTs) for higher order ordinary differential equations (ODEs). This algorithm is based on the idea of seeking transformations that preserve the Painlevé property, and is applied here to ODEs of various orders in order to recover, amongst others, their auto-BTs. Of the ODEs considered here, one is seen to be of particular interest because it allows us to show that auto-BTs can be obtained in various ways, i.e. not only by using the severest of the possible restrictions of our algorithm.     

Special functions as solutions to discrete Painlevé equations

We present results on special solutions of discrete Painlevé equations. These solutions exist only when one constraint among the parameters of the equation is satisfied and are obtained through the solutions of linear second-order (discrete) equations. These linear equations define the discrete analogues of special functions.     

The Painlevé integrability of two parameterized nonlinear evolution equations using symbolic computation

An algorithm is presented to prove the Painlevé integrability of parameterized nonlinear evolution equations such that one can filter out Painlevé integrable models from nonlinear equations with general forms. Then two well known nonlinear models with physical interests illustrate the effectiveness of this algorithm. Some new results are reported for the first time.     

Correlation between pole location and asymptotic behavior for Painlevé I solutions

We extend the technique of asymptotic series matching to exponential asymptotics expansions (transseries) and show by using asymptotic information that the extension provides a method of finding singularities of solutions of nonlinear differential equations. This transasymptotic matching method is applied to Painlevé's first equation, P1. The solutions of P1 that are bounded in some direction towards infinity can be expressed as series of functions obtained by generalized Borel summation of formal transseries solutions; the series converge in a neighborhood of infinity. We prove (under certain restrictions) that the boundary of the region of convergence contains actual poles of the associated solution. As a consequence, the position of these exterior poles is derived from asymptotic data. In particular, we prove that the location of the outermost pole x_p(C) on ℝ⁺ of a solution is monotonic in a parameter C describing its asymptotics on anti‐Stokes lines and obtain rigorous bounds for x_p(C). We also derive the behavior of x_p(C) for large C ∈ ℂ. The appendix gives a detailed classical proof that the only singularities of solutions of P1 are poles. © 1999 John Wiley & Sons, Inc.     

Value distribution of the fifth Painlevé transcendents in sectorial domains

This article is concerned with a value distribution of the fifth Painlevé transcendents in sectorial domains around a fixed singular point. We show that the cardinality of the 1-points of a fifth Painlevé transcendent in a sector has an asymptotic growth of finite order, thereby giving an improvement of the known estimates.     

Painlevé IV with both Parameters Zero: A Numerical Study

The six Painlevé equations were introduced over a century ago, motivated by rather theoretical considerations. Over the last several decades, these equations and their solutions, known as the Painlevé transcendents, have been found to play an increasingly central role in numerous areas of mathematical physics. Due to extensive dense pole fields in the complex plane, their numerical evaluation remained challenging until the recent introduction of a fast “pole field solver” [ 1 ]. The fourth Painlevé equation has two free parameters in its coefficients, as well as two free initial conditions. The present study applies this new computational tool to the special case when both of its parameters are zero. We confirm existing analytic and asymptotic knowledge about the equation, and also explore solution regimes which have not been described in the previous literature.     

Cubic and Quartic Transformations of the Sixth Painlevé Equation in Terms of Riemann–Hilbert Correspondence

The starting point of this paper is a classification of quadratic polynomial transformations of the monodromy manifold for the 2 × 2 isomonodromic Fuchsian systems associated to the Painlevé VI equation. Up to birational automorphisms of the monodromy manifold, we find three transformations. Two of them are identified as the action of known quadratic or quartic transformations of the Painlevé VI equation. The third transformation of the monodromy manifold gives a new transformation of degree 3 of Picard’s solutions of Painlevé VI.     

A (3+1)‐dimensional Painlevé integrable model obtained by deformation

To find some non‐trivial higher‐dimensional integrable models (especially in (3+1) dimensions) is one of the most important problems in non‐linear physics. An efficient deformation method to obtain higher‐dimensional integrable models is proposed. Starting from (2+1)‐dimensional linear wave equation, a (3+1)‐dimensional non‐trivial non‐linear equation is obtained by using a non‐invertible deformation relation. Further, the Painlevé integrability of the resulting model is also proved. Copyright © 2002 John Wiley & Sons, Ltd.     

On a Painlevé II Model in Steady Electrolysis: Application of a Bäcklund Transformation

A model equation of Painlevé II type was introduced by Bass in 1964 in connection with a boundary value problem which describes the electric field distribution in a region x > 0 occupied by an electrolyte. This is possibly the earliest explicit physical application of a Painlevé equation to be found in the literature. Here we return to this problem informed by the subsequent discovery of a Bäcklund transformation for Painlevé II. This enables us to construct exact representations for the electric field and ion distributions for boundary value problems wherein the ratio of fluxes of the positive and negative ions at the boundary adopts one of an infinite sequence of values.     

A Direct Proof that Solutions of the Six Painlevé Equations Have No Movable Singularities Except Poles

The Painlevé property of an nth-order differential equation is that no solution has any movable singularities other than poles. This property is strongly indicative of complete integrability (the existence of n ? 1 integrals). However, the usual technique employed to test for the Painlevé property seeks only movable algebraic (or logarithmic) singularities. More general singularities are ignored. But, the six standard Painlevé equations are known to have no such singularities. Painlevé's proof of this is long and laborious; we give here a direct proof.     

Tanh-travelling wave solutions, truncated Painlevé expansion and reduction of Bullough–Dodd equation to a quadrature in magnetohydrodynamic equilibrium

The equations of magnetohydrodynamic (MHD) equilibria for a plasma in gravitational field are investigated. For equilibria with one ignorable spatial coordinate, the MHD equations are reduced to a single nonlinear elliptic equation for the magnetic potential , known as the Grad–Shafranov equation. Specifying the arbitrary functions in this equation, the Bullough–Dodd equation can be obtained. The truncated Painlevé expansion and reduction of the partial differential equation to a quadrature problem (RQ method) are described and applied to obtain the travelling wave solutions of the Bullough–Dodd equation for the case of isothermal magnetostatic atmosphere, in which the current density J is proportional to the exponentially of the magnetic flux and moreover falls off exponentially with distance vertical to the base, with an “e-folding” distance equal to the gravitational scale height.     

On the Painlevé integrability, periodic wave solutions and soliton solutions of generalized coupled higher-order nonlinear Schrödinger equations

It is proven that generalized coupled higher-order nonlinear Schrödinger equations possess the Painlevé property for two particular choices of parameters, using the Weiss–Tabor–Carnevale method and Kruskal’s simplification. Abundant families of periodic wave solutions are obtained by using the Jacobi elliptic function expansion method with the assistance of symbolic manipulation system, Maple. It is also shown that these solutions exactly degenerate to bright soliton, dark soliton and mixed dark and bright soliton solutions with physical interests.     

A note on the Painlevé analysis of a (2 + 1) dimensional Camassa–Holm equation

We investigate the Painlevé analysis for a (2 + 1) dimensional Camassa–Holm equation. Our results show that it admits only weak Painlevé expansions. This then confirms the limitations of the Painlevé test as a test for complete integrability when applied to non-semilinear partial differential equations.     

On the Jacobi Last Multiplier, integrating factors and the Lagrangian formulation of differential equations of the Painlevé–Gambier classification

We use a formula derived almost seventy years ago by Madhav Rao connecting the Jacobi Last Multiplier of a second-order ordinary differential equation and its Lagrangian and determine the Lagrangians of the Painlevé equations. Indeed this method yields the Lagrangians of many of the equations of the Painlevé–Gambier classification. Using the standard Legendre transformation we deduce the corresponding Hamiltonian functions. While such Hamiltonians are generally of non-standard form, they are found to be constants of motion. On the other hand for second-order equations of the Liénard class we employ a novel transformation to deduce their corresponding Lagrangians. We illustrate some particular cases and determine the conserved quantity (first integral) resulting from the associated Noetherian symmetry. Finally we consider a few systems of second-order ordinary differential equations and deduce their Lagrangians by exploiting again the relation between the Jacobi Last Multiplier and the Lagrangian.