Nonclassical Potential Symmetries of the Burgers Equation

In this paper, new classes of symmetries for partial differential equations (PDE) which can be written in a conserved form are introduced. These new symmetries called nonclassical potential symmetries, are neither potential symmetries nor nonclassical symmetries. Some of these symmetries are carried out for the Burgers equation ut + uux − uxx = 0. (1) by studying the nonclassical symmetries of the integrated equation vt + v x 2 − vxx = 0. (2) By comparing the classical symmetries of the associated system vx = u, vt = ux − u 2 2 (3) with those of the integrated equation (2), we deduce the condition for the symmetries of (2) to yield potential symmetries of (1). The nonclassical potential symmetries are realized as local nonclassical symmetries of (2). Similarity solutions are also discussed in terms of the integrated equation and yield solutions of the Burgers equation which are neither nonclassical solutions of the Burgers equation nor solutions arising from potential symmetries.