Similarity reductions of the KP equation by a direct method

Abstract

Basing on a direct method developed by Clarkson and Kruskal (1989), the Kadomtsev-Petviashvili (KP) equation has been reduced to three types of (1+1)-dimensional partial differential equations which are equivalent to the three types of the similarity reduction equations obtained by the classical Lie approach but with different independent variables. More arbitrary functions which have been missed by the classical Lie approach have been included in the solutions of the KP equation. For instance, the third type of reduction obtained by the direct method can be divided into three subcases and the third type of solution of the KP equation obtained by the classical Lie approach is only a special case of one subcase of the authors' results.