Lie reductions and exact solutions of dispersionless Nizhnik equation

Abstract

We exhaustively classify the Lie reductions of the real dispersionless Nizhnik equation to partial differential equations in two independent variables and to ordinary differential equations. Lie and point symmetries of reduced equations are comprehensively studied, including the analysis of which of them correspond to hidden symmetries of the original equation. If necessary, associated Lie reductions of a nonlinear Lax representation of the dispersionless Nizhnik equation are carried out as well. As a result, we construct wide families of new invariant solutions of this equation in explicit form in terms of elementary, Lambert and hypergeometric functions as well as in parametric or implicit form. We show that Lie reductions to algebraic equations lead to no new solutions of this equation in addition to the constructed ones. Multiplicative separation of variables is used for illustrative construction of non-invariant solutions.