An encyclopedia of Kudryashov’s integrability approaches applicable to optoelectronic devices

Abstract

Purpose:In the last decades, many researchers have performed their best in order to find the solution of the nonlinear evolution equations, nonlinear partial differential equations (NLPDEs), fractional nonlinear partial differential equations and nonlinear optical models by using Kudryashov methods, which have various forms. However, most researchers have used different methods under the same name or applied the same method under the different names. This situation has created confusion for researchers and readers, such as choosing the appropriate method, algorithm and the right article title. In order to eliminate this problem and to clarify the methods, we have conducted rigorous research and wrote a certain standard in this field by bringing together the Kudryashov methods, which are widely used under different names in the literature. Besides, it has been aimed to give the general algorithm of the methods, to make some addendums on the methods and to present new approaches to these methods for the researchers. We strongly believe that the study will be of great benefit not only for those who are interested in nonlinear evolution problems, but also for studies in the field optics. Methodology:Regarding the methods, first of all, the articles of the researchers who suggested the methods and the studies of the respected researchers in this field have been examined and tried to gather under one framework and give a general name with appropriate algorithm for each Kudryashov method. The algorithms and solution functions of each method have been checked on the related Riccati and Bernoulli equations with different softwares (Mathematica, Maple, Matlab, etc.), and confirmed every results carefully. Afterwards, all methods have been applied on (2+1)-dimensional Zoomeron equation for the global test purpose. With the usual standard approach, the nonlinear partial differential form of Zoomeron equation has been transformed into the nonlinear ordinary differential (NODE) form by applying the wave transformation, then each method applied to the NODE form of Zoomeron equation. After these operations, we have obtained a system of linear algebraic equations and their appropriate solutions. By obtaining the appropriate solution sets, the soliton solutions of the Zoomeron equation have been constructed and some selected graphical presentations given. Findings:The results have been obtained from the applications that applied on the Zoomeron equation via Kudryashov methods shows us they can be used widely and effectively in the solution of not only nonlinear evolution problems, but also highly dispersive and higher order nonlinear partial differential equations. It has been seen that all the applied Kudryashov methods give effective results. Besides, controlling and using the parameters in the Kudryashov methods play important role, especially obtaining the effective and desired solution sets. Originality:Based on our detailed research, we could not find any summative study in this area like this one. In addition to all these, some of the approaches and addendums have been presented for the first time in this paper. Besides, (2+1)-dimensional Zoomeron equation has also not been previously solved by some of these methods. It is thought that this study will be used as a reference by many researchers in the future and it will be of great benefit to researchers.