Nonclassical Potential Symmetries and New Explicit Solutions of the Burgers Equation

The symmetry group method is important and widely used in the reduction and construction of explicit solutions of PDEs (partial differential equations). As shown in [1 – 4], the Lie symmetry group method can be used to find explicit solutions of PDEs using symmetry reduction and construction. This method is known as the classical method. By a classical symmetry group of a system of PDEs we mean a continuous group of transformations, which acts on the space of independent and dependent variables and transforms one solution of PDEs into another solution. These solutions are called group-invariant solutions. The first approach to potential symmetries of a system of PDEs was made by Bluman and Cole [5]. In [6], Bluman and Kumei introduced an algorithm which yields new classes of symmetries of given PDEs which are neither Lie point nor Lie-Bäcklund symmetries. They are nonlocal symmetries and are called potential symmetries. In general, the number of determining equations in this kind of symmetry is smaller than in the classical Lie group method. Therefore it is difficult to find all possible solutions of the overdetermined system. Using this new symmetry method, a much wider class of symmetry groups is available. Hence there is the possibility of finding more group-invariant solutions by the same reduction technique. This new symmetry attracts many researchers. Recently, the potential symmetry has been developed and generalized to nonclassical potential symmetries. Several nonclassical potential symmetry gen-