A Study of Lie Point Symmetries, Optimal System, Analytical Solutions and Conversed Vectors of the Mixed Doubly Nonlinear Dispersive Burgers Equation

Burgers equation is a partial differential equation that is a convection-diffusion equation, which arises in many areas of applied mathematics including gas dynamics, nonlinear acoustics, fluid mechanics, and traffic flow.There are several forms of Burgers equations in the literature that are studied by researchers.In this study, using Lie group analysis, we explore a combination of two Burgers hierarchies which is called the mixed doubly nonlinear dispersive Burgers (mDDB) equation.Firstly, we compute the Lie point symmetries of the mDDB equation and then construct the optimal system of one-dimensional subalgebras.Using this optimal system we transform the mDDB equation into several nonlinear ordinary differential equations, which are then solved using various methods including the simplest equation method and the power series expansion method.Furthermore, we present the 3D, density, and 2D graphs of the derived solutions that demonstrate their dynamic behaviours.Finally, the conserved vectors of the mDDB equation are displayed using the multiplier technique and Ibragimov's method.These include the conserved energy and linear momentum.