NONCLASSICAL SYMMETRY REDUCTIONS FOR THE KADOMTSEV-PETVIASHVILI EQUATION

In this paper we discuss dimensional reductions of the physically and mathematically significant Kadomtsev-Petviashvili equation. In particular, we exhibit some new reductions of the Kadomtsev-Petviashvili equation which are obtained by a generalization of the direct method for determining similarity reductions recently developed by Clarkson and Kruskal [J. Math. Phys. 30 (1989) 2201]. New solutions of the Kadomtsev-Petviashvili equation are obtained in terms of Weierstrass elliptic functions, solutions of the Lame equations, and the first, second and fourth Painleve transcendents. A group theoretical explanation of the new solutions is provided in terms of conditional symmetries, leaving only specific subsets of solutions of the Kadomtsev-Petviashvili equation invariant.