Rational Solutions of the Burgers Hierarchy

Abstract

   Rational solutions of the Burgers hierarchy have been obtained using the linearization of the differential equation. Hierarchy equations allow a group of stretch transformations. Using self-similar variables, Burgers equations are converted after integration into an n-order differential equation where the function depends only on one variable. The linearization of the equation is done using the Cole–Hopf transformation. A solution of an ordinary linear differential equation is sought in the form of an (n + 1)-order polynomial. The substitution of the polynomial with indefinite coefficients into the equations transforms the linear differential equation into a system of (n + 1)-order algebraic equations with constant coefficients. The kind of a rational solution depends on the constant of integration, which has certain values related to the degree of polynomial. Rational solutions of the Burgers hierarchy have points of discontinuity.