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

Rational solutions of the second, third and fourth Painlev´e 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 Painlev´e 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 Painlev´e 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 Painlev´e 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 Schr¨odinger equation are expressed in terms of these special polynomials.