Point- and contact-symmetry pseudogroups of dispersionless Nizhnik equation

• Published in: Communications in nonlinear science & numerical simulation Vol. 132; p. 107915
• Main Authors: Boyko, Vyacheslav M., Popovych, Roman O., Vinnichenko, Oleksandra O.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.05.2024
• Subjects: Discrete symmetry; Dispersionless Nizhnik equation; Lie invariance algebra; Point-symmetry pseudogroup; Dispersionless Nizhnik equation; Discrete symmetry; Lie invariance algebra; Point-symmetry pseudogroup
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2024.107915

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. •The point-symmetry pseudogroup G of the dispersionless Nizhnik equation is found.•The contact-symmetry counterpart of G coincides with the first prolongation of G.•This gives the first examples where (pseudo)groups are defined by their algebras.•We describe geometric properties of this equation that completely define it.•The algebraic method of constructing point-symmetry pseudogroups is developed.