Point- and contact-symmetry pseudogroups of dispersionless Nizhnik equation

TL;DR

This paper introduces a novel megaideal-based algebraic method to compute the symmetry pseudogroups of the dispersionless Nizhnik equation, avoiding the direct method and extending to contact symmetries and related equations.

Contribution

It presents the first application of a megaideal-based algebraic approach to symmetry analysis of the dispersionless Nizhnik equation and its related systems, including contact symmetries and geometric properties.

Findings

Computed the point-symmetry pseudogroup without the direct method.

Established that the contact-symmetry pseudogroup coincides with the first prolongation of the point-symmetry pseudogroup.

Identified all third-order PDEs with the same Lie invariance algebra.

Abstract

Applying an original megaideal-based version of the algebraic method, we compute the point-symmetry pseudogroup of the dispersionless (potential symmetric) Nizhnik equation. This is the first example of this kind in the literature, where there is no need to use the direct method for completing the computation. The analogous studies are also carried out for the corresponding nonlinear Lax representation and the dispersionless counterpart of the symmetric Nizhnik system. We also first apply the megaideal-based version of the algebraic method to find the contact-symmetry (pseudo)group of a partial differential equation. It is shown that the contact-symmetry pseudogroup of the dispersionless Nizhnik equation coincides with the first prolongation of its point-symmetry pseudogroup. We check whether the subalgebras of the maximal Lie invariance algebra of the dispersionless Nizhnik equation that naturally arise in the course of the above computations define the diffeomorphisms stabilizing this algebra or its first prolongation. In addition, we construct all the third-order partial differential equations in three independent variables that admit the same Lie invariance algebra. We also find a set of geometric properties of the dispersionless Nizhnik equation that exhaustively defines it.