Symmetry reduction and exact solutions of the generalized Nizhnik-Novikov-Veselov equation

Combining the generalized direct method with the classical Lie method, we investigate the generalized Nizhnik-Novikov-Veselov (GNNV) equation. First, by means of the generalized direct method, a relationship is constructed between the new solutions and the old ones of the equation. At the same time, we obtain the full symmetry group of the GNNV equation, which includes the Lie point symmetry group S and the discrete groups D. By using the symmetry, the (2+1)-dimensional GNNV equations are reduced to (1+1)-dimensional reduced equations. Second, the symmetry of the reduced equation is obtained via the classical Lie method. Then we reduce the reduced equation to the ordinary differential equations (ODEs) using the symmetry. Solving the ODEs and based on the relationship obtained above, new solutions of the GNNV equation are obtained. At last, the more general solutions are obtained, which recover one of the solutions by Radha and Lakshmanan [R. Radha, M. Lakshmanan, J. Math. Phys. 35 (9) (1994) 4746].