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

Abstract

By using the new recently proposed extended rational methods, we set some solitary and multi wave solutions of the coupled Higgs system in the present study. Some new families of hyperbolic and trigonometric solutions of the coupled Higgs equations are successfully obtained. Some solutions are real as some are complex valued functions. Particular forms of some solutions derived by choice of some parameters are demonstrated in three dimensional spaces. The complex valued solutions are also depicted by plotting both real and complex component.

Traveling wave and general form solutions for the coupled Higgs system

“…Traveling wave solutions obtained with the help of the applied method have been known as analytical localized solutions 57 . Traveling wave solutions obtained in this study as a result of the application of the method are different from the literature 42–51 . In addition, the solutions presented with Equations () and () as a result of the specific definition of three parameters, namely, wave number, wave velocity, and frequency of the soliton, have been defined as general solutions.…”

“…57 Traveling wave solutions obtained in this study as a result of the application of the method are different from the literature. [42][43][44][45][46][47][48][49][50][51] In addition, the solutions presented with Equations (4.1) and (4.2) as a result of the specific definition of three parameters, namely, wave number, wave velocity, and frequency of the soliton, have been defined as general solutions. These solutions are more general than traveling wave solutions and differ from the literature.…”

“…In 2018, solitary wave solutions were produced by Khater for Equation () with the help of the generalized Exp‐function method 42 . In 2018, traveling wave solutions were obtained by Rezazadeh et al for CHFE using new extended rational methods 43 . In 2020, localized wave solutions for CHFE were created using the symbolic computation method by Zhaqilao 44 .…”

“…42 In 2018, traveling wave solutions were obtained by Rezazadeh et al for CHFE using new extended rational methods. 43 In 2020, localized wave solutions for CHFE were created using the symbolic computation method by Zhaqilao. 44 Some of these solutions are called higher order rogue wave, the Ma breather, the Akhmediev breather, and the general breather.…”

Exploring cone-shaped solitons, breather, and lump-forms solutions using the lie symmetry method and unified approach to a coupled breaking soliton model

“…Numerous methods have been developed by researchers and scholars to extract solutions for different types of NLPDEs [1][2][3]. These methods include the Painlevé test [4], the F-expansion method [5], the Lie symmetry method [6,7], the Bell polynomial method [8], the generalized exponential rational function method [9], the generalized Riccati equation mapping method [10], the Hirota bilinear method [11], the Backlund transformation [12], the Exp-function method [13], geometric approach [14], the extended tanh-coth expansion method [15], the Extended Sinh-Gordon equation method [16], the extended trial equation scheme [17], the Darboux transformation method [18,19], the generalized perturbation (n, N − n)-fold Darboux transformation [20], the Extended rational sin-cos method [21] and many more [22]. These various methods offer valuable tools for solving the complex problems presented by NLPDEs.…”