New Exact and Explicit Travelling Wave Solutions for the Coupled Higgs Equation and a Nonlinear Variant of the PHI-four Equation

Abstract

An analytic study on a complex nonlinear system (Coupled Higgs equation) and a nonlinear variant of the PHI-four equation is presented in this paper. The Exp-function method is employed to derive exact periodic and generalized solitary solutions of these equations. The solutions are compared with those obtained by the tanh method, the sine-cosine method and the Weierstrass elliptic function method, furthermore, new and more general solutions are found.

Traveling wave and general form solutions for the coupled Higgs system

“…We can say that the traveling wave solutions obtained in this study are different and more general than the traveling wave solutions obtained by Khajeh et al, 38 which we discussed in the introduction. This situation constitutes one of the original values of our study.…”

“…These solutions are more general than traveling wave solutions and differ from the literature. 38 It was observed that the solutions obtained to test the reliability of the MGERFM provided the CHS after symbolic calculations. In addition, in the results and discussion section, the behaviors exhibited by the traveling wave for different values of the soliton frequency and wave number in Equation (3.8) have been presented as a simulation in Figure 10, and the physical interpretations of parameters have been discussed in detail.…”

“…Second, these solutions have two local maximum families, and finally, their solutions are investigated by associating them with equations in different forms with the help of transformations. In 2010, Khajey et al reached generalized solitary solutions for Equation () with the help of the Exp‐function method 38 . As it is known in 38 references, giving the solutions in general form constituted the most distinctive feature that distinguishes this study from other studies in the literature.…”

“…In 2010, Khajey et al reached generalized solitary solutions for Equation () with the help of the Exp‐function method 38 . As it is known in 38 references, giving the solutions in general form constituted the most distinctive feature that distinguishes this study from other studies in the literature. In 2011, Tang and Xia handled and studied all traveling wave solutions connected in parameter space with the help of the bifurcation theory of dynamical systems 39 .…”

Extended Rational Sinh-Cosh and Sin-Cos Methods to Derive Solutions to the Coupled Higgs System

“…The conditions to guarantee the existence of these solutions were also summarized in the same study. Various solutions in hyperbolic or trigonometric function forms were determined by implementation of exponential function technique [6]. Jacobi type periodic, hyperbolic and trigonometric solutions in wave forms were set by an algebraic method [7].…”