On Nonlinear Monge-Ampère Equations and Their Symmetry Classifications

Abstract

The Monge-Ampère equation (MAE) plays a pivotal role across a broad spectrum of theoretical and applied sciences, with its solutions being essential for advancing various fields. This study explores all forms of the fully nonlinear MAE using Lie group transformations to reduce the equation into solvable forms. Analytical solutions are derived using ansatzbased methods, yielding novel and generalized results that enhance the existing body of knowledge. In particular, solutions for cases with diverse source functions and boundary conditions are obtained, addressing gaps in the literature. Stability of the solutions is studied through both analytical and numerical approaches. Comparisons with existing solutions demonstrate the efficiency and generality of the proposed methods. The results presented in this work are poised to impact numerous applications, providing a robust framework for further research on MAEs.OPEN ACCESS Received: 24/05/2025 Accepted: 07/08/2025 Published: 27/10/2025

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