A Note on Exact Travelling Wave Solutions for the Klein–Gordon– Zakharov Equations

Abstract

In this paper, we investigate the travelling wave solutions for the Klein-Gordon-Zakharov equations by using the modified trigonometric function series method benefited to the ideas of Z. Y. Zhang, Y. X. Li, Z. H. Liu, and X. J. Miao, Commun. Nonlin. Sci. Numer. Simul. 16, 3097 (2011). Exact travelling wave solutions are obtained.

Traveling Wave Solutions for Nonlinear Differential-Difference Equations of Rational Types

“…Indeed, the following well-known identities sinh(x ± y) = sinh x cosh y ± cosh x sinh y, cosh(x ± y) = cosh x cosh y ± sinh x sinh y, sin(x ± y) = sin x cos y ± cos x sin y, cos(x ± y) = cos x cos y ∓ sin x sin y, and the Lemma along with the identity ξ n±p s = ξ n ± ϕ s provides (22). Then, substituting ( 22) into ( 17) gives (24).…”

The Global Existence and Uniqueness of the Classical Solution with the Periodic Initial Value Problem for One-Dimension Klein-Gordon-Zakharov Equations

“…Besides, in [10], by combining the trigonometric function series method and the exp-function method, Y. Zhang et al considered the travelling wave solutions of (7). Now, let us describe our results.…”

Construction of optical solitons and other families of solutions to the nonlinear Biswas-Arshed model in birefringent fibers without four-wave mixing terms

“…In the past few decades, many researchers are interested in studying the nonlinear Shrödinger (NLS) equation as a nonlinear PDE, which has many applications in optical fibers, plasma, and other areas of sciences and engineering [1][2][3]. Many approaches leading to exact solutions of nonlinear PDEs in the literature, such as the Hirota's bilinear [4,5], the trigonometric function series method [6], the modified mapping method [7], the modified trigonometric function series method [8,9], the bifurcation method [10,11], the tanh-coth method [12], the Jacobi elliptic function method [13,14], the exp-function method [15], the F-expansion method [16], the mapping method [17], new φ 6 -model expansion method [18], unified Riccati equation expansion method [19], modified simple equation method [20], extended simplest equation method [20,21], generalized Sub-ODE method [22][23][24], the new extended auxiliary equation method [25,26] and others. The idea in solving nonlinear PDE via most of these techniques is to reduce it to a nonlinear ordinary differential equation (ODE) and hence solving it by the procedures of these approaches leading to the exact solutions to the original PDE under consideration.…”