Analysis of the symmetry group and exact solutions of the dispersionless KP equation in $n+1$ dimensions

TL;DR

This paper analyzes the symmetry group of the (n+1)-dimensional dispersionless KP equation, classifies its Lie algebra, and shows how known solutions are invariant under specific symmetry transformations.

Contribution

It identifies the Lie algebra structure of the symmetry group and demonstrates the group-invariance of known solutions for the dispersionless KP equation.

Findings

The symmetry algebra is a semi-direct sum of a simple Lie algebra and a nilpotent subalgebra.

Solutions are shown to be group-invariant under the $ ext{SL}(2, r)$ subgroup.

Explicit solutions correspond to specific infinitesimal generators of the symmetry group.

Abstract

The Lie algebra of the symmetry group of the $(n+1)$-dimensional ge\-ne\-ra\-li\-zation of the dispersionless Kadomtsev--Petviashvili (dKP) equation is obtained and identified as a semi-direct sum of a finite dimensional simple Lie algebra and an infinite dimensional nilpotent subalgebra. Group transformation properties of solutions under the subalgebra $\Sl(2,\mathbb{R})$ are presented. Known explicit analytic solutions in the literature are shown to be actually group-invariant solutions corresponding to certain specific infinitesimal generators of the symmetry group.