Traveling-wave solution of the Tzitzéica-type equations by using the unified method

The Tzitzéica equation, the Dodd–Bullough–Mikhailov equation, and the Tzitzéica–Dodd–Bullough equation arise in various branches of science and technology, such as solid state physics, nonlinear dynamics, nonlinear optics, and quantum field theory problems. In this paper, we discuss how to find more general wave solutions of these three nonlinear evolution equations with physical applications using the unified method. The unified method straightforwardly gives many more general solutions with free parameters without the need for extra hardware support. After performing calculations, the graphs of the solutions are plotted using Maple to give insight into the physical structure of the wave solution. Only some selected solutions are plotted to visualize their behavior. Considering the importance of having solutions for nonlinear wave phenomena, we clearly see that the method has many merits and demonstrates a comprehensive application to obtaining solutions in an efficient way. In addition, diverse types of geometrically structured solitons such as an anti-bell-shaped soliton, a flat soliton, a kink, and a singular soliton are produced by using arbitrary parameters.