An analysis of symmetries and conservation laws of nonlinear partial differential equations arising from Burgers’ hierarchy

Abstract

We investigate the nonlinear evolutionary partial differential equations (PDEs) derived from Burgers’ hierarchy and give the exact solution of the complete hierarchy. The conservation laws of the hierarchy are studied and we proceed to establish the general nth conservation law. A transformation is used to render the hierarchy to a hierarchy of nonlinear ordinary differential equations (ODEs). These expressions are then linearised. Ultimately we give a novel exact solution of the entire Burgers’ hierarchy, that is, for all values of n. Several members of the hierarchy are solved, and the graphical counterparts of their solutions are provided to illustrate the applicability of our formula. Next we extend our study to the hierarchy of ODEs linked to this hierarchy. One-parameter Lie group of transformations that leave the ODEs invariant are constructed, from which it is established that these symmetries arise from the (n+ 1) complex roots of a certain polynomial. This gives us a formula to solve the ODE expressions, and finally we show how a more general exact solution of the complete hierarchy is obtained from this result