LIE GROUP STRUCTURES AND SOLITON SOLUTIONS TO THE NONLINEAR SCHRO¨DINGER EQUATION

Abstract

This paper investigates the Lie symmetry structure, conservation laws, and novel soliton solutions of a nonlinear Schro & uml;dinger equation. We determine the admitted Lie point symmetries and establish the associated Lie algebra generated by Gamma(1 )= partial derivative(x), Gamma(2) = partial derivative(t), and Gamma(3) = x partial derivative(x) +2t partial derivative(t). The algebraic structure is explored through the commutator table, adjoint representation, and classification of the optimal system of subalgebras. Corresponding similarity reductions transform the PDE system into reduced ordinary differential equations, for which approximate solutions are constructed using power series methods. In addition, conservation laws are systematically derived via conservation theorem, linking the symmetries to physically meaningful invariants. Novel soliton solutions have been established using the modified generalized Riccati equation mapping method (MGREMM). The results provide a comprehensive symmetry-based framework for the system under study, contributing to the broader understanding of nonlinear coupled PDEs and their analytical properties.