Lie symmetry analysis, optimal system, conservation laws, and exact solutions for the variable-coefficient higher-order Boussinesq–Burgers system

In this paper, we present a comprehensive examination of the Lie symmetry analysis, optimal system, exact solutions, and conservation laws pertaining to the variable-coefficient higher-order Boussinesq–Burgers system. The Lie symmetry analysis method is initially employed to derive the vector field, followed by the utilization of Olver’s method to determine the optimal system of one-dimensional subalgebras for the system. By utilizing the optimal system, the Boussinesq–Burgers system is similarly reduced. Subsequently, the [Formula: see text]-expansion method is applied to this reduced system, yielding kink soliton solution, dark soliton solution, trigonometric-function solution and rational solution for the variable-coefficient higher-order Boussinesq–Burgers system. By assigning various functions to the coefficient functions, different types of soliton solutions are obtained, and these solutions’ structural features are visualized through the use of three-dimensional figures. Last, an examination of the conservation laws governing the system is conducted.