Generalized Hermite polynomials for the Burgers hierarchy and point vortices

Abstract

Rational solutions of equations for the Burgers hierarchy are considered. Using self-similar variables this hierarchy is reduced to the family of nonlinear ordinary differential equations. Then the family is transformed to the hierarchy of non-autonomous linear differential equations by means of the Cole-Hopf formula. This hierarchy is a generalization of the second-order equation for Hermite polynomials. It is shown that every member of the hierarchy for ordinary differential equation has the solution in the form of polynomials. Properties of solutions of generalized Hermite equations in the form the special polynomials are studied. A recursion relation that can be used for finding corresponding polynomials for every member is given. It is proved that the well-known property for Hermite polynomials connecting two polynomials can be used for all polynomials of the generalized Hermite hierarchy. It is shown that the Cole-Hopf transformation is a direct consequence of the differential connection between two special polynomials in the hierarchy of Hermite equations. A derivation of the generalized Tkachenko equations is given for polynomials of the generalized Hermite hierarchy whose roots correspond to point vortices in the background flow.