Similarity and conditional similarity reductions of a (2+1)-dimensional KdV equation via a direct method

To get the similarity solutions of a nonlinear physical equation, one may use the classical Lie group approach, nonclassical Lie group approach and the Clarkson and Kruskal (CK) direct method. In this paper the direct method is modified to get the similarity and conditional similarity reductions of a (2 + 1) dimensional KdV-type equation. Ten types of usual similarity reductions [including the (1 + 1)-dimensional shallow water wave equation and the variable KdV equation] and six types of conditional similarity reductions of the (2 + 1)-dimensional KdV equation are obtained. Some special solutions of the conditional similarity reduction equations are found to show the nontriviality of the conditional similarity reduction approach. The conditional similarity solutions cannot be obtained by using the nonclassical Lie group approach in its present form. How to modify the nonclassical Lie group approach to obtain the conditional similarity solutions is still open.