Some Traveling Wave Solutions to the Fifth-Order Nonlinear Wave Equation Using Three Techniques: Bernoulli Sub-ODE, Modified Auxiliary Equation, and G ′ / G -Expansion Methods

Abstract

The fifth-order nonlinear wave equation contains terms involving higher-order spatial derivatives, such as $u_{xxx}$ and $u_{xxxxx}$ . These terms are responsible for dispersion, which affects the shape and propagation of the wave. The study of dispersion is important in many areas, including seismology, acoustics, and communication theory. In the current work, three potent analytical techniques are proposed in order to solve the fifth-order nonlinear wave equation. The used approaches are the modified auxiliary equation method, the Bernoulli Sub-ODE method, and the $\left(G\prime /G\right)$ -expansion method (MAE). Some graphs are plotted to display our findings. The solutions to the nonlinear wave equation are used to describe the nonlinear dynamics of waves in physical systems. The results show how the dynamics of the wave solutions are influenced by the system parameters, which can be used as system controllers. The new approaches used in this work helped to find new solutions for traveling waves. This could be seen as a new contribution to the field. Water waves, plasma waves, and acoustic wave behavior can be described by the obtained solutions.