New exact solutions of Burgers’s equation—an extension to the direct method of Clarkson and Kruskal

Abstract

New exact solutions of Burgers’s equation, which is the simplest evolution equation to embody nonlinearity and dissipation, are obtained. These solutions are obtained by extending the method for finding symmetry reductions due to Clarkson and Kruskal [‘‘New similarity solutions of the Boussinesq equation,’’ J. Math. Phys. 30, 2201–2213 (1989)]. The novel feature of this extended method is that one can seek reductions to a system of ordinary differential equations, rather than the usual single equation and this leads to a wider class of solutions. One is able to complete the calculations necessary for Burgers’s equation in all but a single case (where it is necessary to integrate an Abel equation of the second kind). This indicates that the method is practicable. In contrast it is often the case that new methods are of limited use in practice. In particular, solutions in terms of a class of confluent-hypergeometric functions are computed. By comparison, solutions found by the original method due to Clarkson and Kruskal, and other reduction methods, are in terms of parabolic-cylinder functions or Airy functions, which are special cases of this class.