Bäcklund Transformations of Painlevé Equations and Their Applications

Abstract

The Painleve equations (P 1)–(P 6) were first derived around the turn of the century in an investigation by Painleve 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 Painleve property. Although first discovered from strictly mathematical considerations, the Painleve equations have appeared in various physical applications. There has been considerable interest in Painleve 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 Painleve equations may also be thought of as nonlinear analogues of the classical special functions. There is currently much interest in studying the Painleve equations using the isomonodromy deformation method. In this approach, the Painleve equation is written as an integrability condition of a linear system.