paper Traveling wave dynamics in higher-order nonlinear Klein-Gordon equations

Abstract

In this work, we investigate exact traveling wave solutions of higher-order generalized nonlinear Klein-Gordon equations using the cent 6-expansion method. Such nonlinear partial differential equations arise in various physical contexts, including fluid dynamics, nonlinear optics, and quantum field theory. The study focuses on obtaining analytical solutions for specific cases with nonlinearity indices n = 1, 2, 3, and 5, corresponding to physically relevant scenarios. The solutions obtained are generated under specific constraint conditions derived from the parameter relationships and coefficient balance conditions that emerged during the application of the method. These constraints enable the fulfillment of reduced algebraic equations and guarantee the mathematical validity and physical applicability of the obtained wave solutions. Among these solutions, some solutions are expressed in terms of hyperbolic functions and exhibiting singularities at certain points represent wave structures that can be classified in the literature as singular hyperbolic traveling waves, trigonometric traveling waves, and singular periodic traveling wave solutions. Each solution is evaluated from a physical perspective, taking into account the relationships between the parameters in the model, and the behavior of the solution was examined for different values of the wave propagation parameter. The cent 6-method enables the derivation of compactons, solitons, solitary wave patterns, and periodic waveforms in a straightforward and efficient manner. Comparisons with previous solution techniques, such as the tanh-function method and the (G '/G)-expansion method, highlight the efficiency and elegance of the proposed approach. The results contribute to the broader understanding of nonlinear wave phenomena and provide analytical tools for further applications in theoretical and applied physics.