Abstract
The nonlinear wave equation is a significant concern to describe wave behavior and structures. Various mathematical models related to the wave phenomenon have been introduced and extensively being studied due to the complexity of wave behaviors. In the present work, a mathematical model to obtain the solution of the nonlinear wave by coupling the classical Camassa-Holm equation and the Rosenau-RLW-Kawahara equation with the dual term of nonlinearities is proposed. The solution properties are analytically derived. The new model still satisfies the fundamental energy conservative property as the original models. We then apply the energy method to prove the well-posedness of the model under the solitary wave hypothesis. Some categories of exact solitary wave solutions of the model are described by using the Ansatz method. In addition, we found that the dual term of nonlinearity is essential to obtain the class of analytic solution. Besides, we provide some graphical representations to illustrate the behavior of the traveling wave solutions.
Highlights
In the study of nonlinear wave phenomena, the nonlinear partial differential equations are one of the great mathematical models to investigate the problems
The various phenomena of shallow-water waves are led by nonlinear partial differential equations such as the Korteweg-de Vries (KdV) equation [1,2,3,4,5,6,7], the Benjamin-Bona-Mahony (BBM) equation [8,9,10,11], the Symmetric Regularized Long Wave (SRLW) equation [12,13,14,15], the Kawahara equation [16,17,18,19], and the Rosenau equation [20,21,22,23]
We successfully studied the nonlinear wave equation by coupling the classical Camassa-Holm equation and the Rosenau-RLW-Kawahara equation in the case of asymptotic boundary conditions
Summary
In the study of nonlinear wave phenomena, the nonlinear partial differential equations are one of the great mathematical models to investigate the problems. For further understanding of nonlinear behaviors of shallow-water waves, the generalized Rosenau-RLW equation was introduced in the following: ut − uxxt + uxxxxt + ux + βupux = 0, ð1Þ where p ≥ 1 and β are a constant. Biswas et al [31] obtained the solitary solution and two invariance of the generalized Rosenau-Kawahara equation with power law nonlinearity. By adding a viscous term −uxxt into the RosenauKawahara equation, which is called the Rosenau-RLWKawahara equation, ut − uxxt + uxxxxt + ux + uxxx − uxxxxx + βupux = 0: ð2Þ It has been a growing interest in computation nonlinear wave equations. By using the sech and trigonometric function method, Esfahani [58] (Esfahani and Pourgholi [59]) studied solitary wave solutions to the generalized Rosenau-KdV and Rosenau-RLW equation, respectively.
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