Auto Backlund transformation, Lie Symmetry Analysis, and Diverse Soliton Structures of an Extended Fifth-Order Nonlinear Evolution Equation

This paper investigates the integrability and exact solutions of an extended fifth-order nonlinear evolution equation. The Painlevé test is applied to confirm the integrability of the model. Exact solutions are then constructed using a combination of symmetry analysis and direct analytical methods. In particular, Lie symmetry reductions are employed to transform the governing partial differential equation into ordinary differential equations, which are solved using the generalized Kudryashov and Kumar–Malik methods. The obtained solutions include bright and dark solitons, kink-type waves, and periodic structures, expressed in terms of hyperbolic and trigonometric functions. Graphical illustrations are presented to demonstrate the physical behavior of these solutions. The results reveal that the model supports a rich variety of nonlinear wave structures, contributing to the understanding of soliton dynamics in higher-order nonlinear systems.