Lie reductions and exact solutions of dispersionless Nizhnik equation

TL;DR

This paper classifies all Lie reductions of the dispersionless Nizhnik equation, analyzes symmetries, and constructs new explicit solutions using various mathematical functions, enhancing understanding of its solution space.

Contribution

It provides a comprehensive classification of Lie reductions and constructs a wide range of explicit solutions, including invariant and non-invariant types, for the dispersionless Nizhnik equation.

Findings

Complete classification of Lie reductions to PDEs and ODEs.

Construction of new explicit solutions in elementary, Lambert, and hypergeometric functions.

Identification that algebraic reductions do not yield new solutions.

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.