Special Polynomials Associated with Rational Solutions of the Painlevé Equations and Applications to Soliton Equations

Abstract

Rational solutions of the second, third and fourth Painleve equations can be expressed in terms of special polynomials defined through second order bilinear differential-difference equations which are equivalent to the Toda equation. In this paper the structure of the roots of these special polynomials, as well as the special polynomials associated with algebraic solutions of the third and fifth Painleve equations and equations in the PII hierarchy, are studied. It is shown that the roots of these polynomials have an intriguing, highly symmetric and regular structure in the complex plane. Further, using the Hamiltonian theory for the Painleve equations, other properties of these special polynomials are studied. Soliton equations, which are solvable by the inverse scattering method, are known to have symmetry reductions which reduce them to Painleve equations. Using this relationship, rational solutions of the Korteweg-de Vries and modified Korteweg-de Vries equations and rational and rational-oscillatory solutions of the non-linear Schrodinger equation are expressed in terms of these special polynomials.