Symmetry reduction for the Kadomtsev–Petviashvili equation using a loop algebra

Abstract

The Kadomtsev–Petviashvili (KP) equation (ut+3uux/2+ 1/4 uxxx)x +3σuyy/4=0 allows an infinite-dimensional Lie group of symmetries, i.e., a group transforming solutions amongst each other. The Lie algebra of this symmetry group depends on three arbitrary functions of time ‘‘t’’ and is shown to be related to a subalgebra of the loop algebra A(1)4. Low-dimensional subalgebras of the symmetry algebra are identified, specifically all those of dimension n≤3, and also a physically important six-dimensional Lie algebra containing translations, dilations, Galilei transformations, and ‘‘quasirotations.’’ New solutions of the KP equation are obtained by symmetry reduction, using the one-dimensional subalgebras of the symmetry algebra. These solutions contain up to three arbitrary functions of t.