Lie Group Classification, Symmetry Reductions, and Conservation Laws of a Monge–Ampère Equation

The nonhomogeneous Monge–Ampère equation, read as wxxwyy−wxy2+h(w)=0, is a nonlinear equation involving mixed second derivatives with respect to the spatial variables x and y, along with an additional source function h(w). This equation is observed in several fields, including differential geometry, fluid dynamics, and magnetohydrodynamics. In this study, the Lie symmetry method is used to obtain a detailed classification of this equation. Symmetry analysis leads to a comprehensive classification of the equation, resulting in specific forms of the smooth source function h(w). Furthermore, the one-dimensional optimal system of the associated Lie algebras is derived, allowing for symmetry reductions that yield several exact invariant solutions of the Monge–Ampère equation. In addition, conservation laws are constructed using the Noether approach, a highly effective and widely used method for deriving conserved quantities. These conservation laws can help evaluate the accuracy and reliability of numerical methods.