Differential invariants of systems of two nonlinear elliptic partial differential equations by Lie symmetry method

The Lie symmetry method offers a systematic approach for analyzing and solving differential equations by identifying continuous transformations that preserve their structure. In this study, we investigate a general system of two nonlinear second-order elliptic partial differential equations using Lie symmetry techniques. We compute the equivalence transformations for the system, which serve as the foundation for deriving differential invariants. Specifically, we establish both joint differential invariants that are obtained under transformations of dependent and independent variables along with semi-differential invariants, derived solely from transformations of dependent variables. These invariants play a crucial role in reducing the system to its simplest possible form while retaining its essential features. By applying these differential invariants, we present reduced forms of various nonlinear systems of elliptic partial differential equations, demonstrating the effectiveness of the method in simplifying complex equations. Our results highlight the utility of Lie symmetry analysis in deriving invariant structures and facilitating the systematic reduction of coupled nonlinear systems of partial differential equations.