Bäcklund Transformations of Painlevé Equations and Their Applications

Abstract

The Painlevé equations (P ₁)–(P ₆) were first derived around the turn of the century in an investigation by Painlevé and his colleagues. Nonlinear ordinary differential equations have the property that the singularities other than poles of any of the solutions are independent of the particular solution and so are independent only of the equation; this property is known as the Painlevé property. Although first discovered from strictly mathematical considerations, the Painlevé equations have appeared in various physical applications. There has been considerable interest in Painlevé equations over the last few years due the fact that they arise as reductions of solutions of soliton equations solvable by the inverse scattering method. The Painlevé equations may also be thought of as nonlinear analogues of the classical special functions. There is currently much interest in studying the Painlevé equations using the isomonodromy deformation method. In this approach, the Painlevé equation is written as an integrability condition of a linear system.

There have been many recent studies of integrable mappings and discrete systems, including discrete analogues of the Painlevé equations. Therefore, any success in the study of Painlevé equations provides new results in the associated problems.

In this chapter we shall discuss the recurrence relations and their application to the construction of various kinds of solutions of the Painlevé equations. In the context of the Painlevé equations, recurrence relations are usually referred to as Bäcklund transformations. Informally, a Bäcklund transformation is defined as a system of equations relating one solution of a given equation either to another solution of the same equation, possibly with different values of the parameters, or to a solution of another equation. In this approach one can derive various properties of the Painlevé equations, including exact solutions (rational, algebraic, classical), the fundamental domain of the parameter space, special integrals, and transcendence. This is the purpose of the present chapter.

It is organized as follows. In the introduction we discuss the Painlevé property and Bäcklund transformations. In Section 2 we review several impornt general properties of the second Painlevé equation (P ₂), such as the character of the singularities, the representation of the solutions as a ratio of entire functions, T-functions, the number of poles of the solutions. Then from the Hamiltonian system we obtain differential and integral forms of Bäcklund transformations of (P ₂). Using this result we obtain algebraic nonlinear superposition of solutions (Bianchi formula or a discrete analogue of (P ₂)) and the fundamental domain of the parameter space. Since one of the principal aims of our investigation is to obtain solution hierarchies for the Painlevé equations by making multiple applications of Bäcklund transformations, in Sections 3 and 4 we obtain the rational solutions and classical solutions that can be expressed in terms of Airy functions. Then in Section 5 we discuss the transcendence of (P ₂).

In the sixth section we consider the higher analogue of (P ₂) that arises as an exact reduction of the higher analogue of the Korteweg-de Vries equation . We show that the equations of the higher analogue have the same properties of solutions as (P ₂). In Sections 7–9 we give a brief discussion of elementary properties of (P ₄) and construction of a Bäcklund transformation of (P ₄). Then we describe the effect on the parameters of (P ₄) of repeated application of Bäcklundtransformation Then we apply these results to finding the parameter criterion of rational solutions and solutions that can be expressed by means ofsolutions of the Weber-Hermite equation. This family includes as specialcases solutions expressible in terms of the error function. Then we obtainthe Weyl chamber in parameter space and discuss the transcendence of (P ₄).

In Section 10 we review several important properties of the third Painlevéequation (P ₃), and include here a number of elementary scaling transformations, the first integrals, and the special Riccati equation. With the exception of the cases where (P ₃) is fully integrable (α = γ = 0 or β = δ = 0), we are able to restrict our attention to two specialized cases, i.e., γδ ≠ 0 (with no loss of generality γ = 1, δ = - 1) and δ = 0, αδ ≠ 0 (α = 1, δ = -1), which simplifies our ensuing discussion.

In Sections 10-13 we obtain the Bäcklund transformations for the two considered cases of (P ₃). This information characterizes the parameter sets for which exact solutions exist. We conduct a systematic study of these solutions and illustrate how they can be categorized into three hierarchies; one of solutions rational in z, a second of algebraic solutions that are rational in z1/3, and finally a family whose members can be expressed in terms of suitable Bessel functions.

In Sections 12, 14 we show the reducibility of equation (P ₃) to (P ₅) and back for some values of the parameters. Then we obtain the fundamental domain and discuss the transcendence of (P ₃). Finally, in Sections 14, 15 we give a brief consideration of integrability of (P ₅) and (P6), Bäcklund transformations, and solution hierarchies of these equations.