Preliminary Group Classification and Some Exact Solutions of the 2-Hessian Equation

Abstract

This paper systematically investigates the Lie symmetry analysis of a class of 3-dimensional non-linear 2-Hessian equations $$u_{xx}u_{yy}+u_{xx}u_{yy}+u_{yy}u_{zz}=u_{xy}^2+u_{yz}^2+u_{xz}^2+f$$ , where f is an arbitrary smooth function f of the variables (x, y, z). In fact, the preliminary group classification of the 2-Hessian equation was carried out. That means, we find one-dimensional Lie symmetry extensions of the principal 4-dimensional sub-algebra of the equivalence algebra of these equations. So, we find additional equivalence transformation on the space (x, y, z, f), with the aid of Bila’s method, and, we take their projections on this space. Then we obtain an optimal system of one-dimensional Lie sub-algebras of these equations which are generated by some vectors and presented on Theorem (5.1). Some new non-linear invariant models are obtained which have non-trivial invariance algebras. The results of the preliminary group classification are some inequivalent equations which summarized in a table. Finally, some exact solutions of the $$2-$$ Hessian equation are presented, and some figures for the obtained solutions are depicted.