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The fundamental objective of this work is focused to achieve a class of advanced and impressive exact estimations to the Zoomeron equation and the time-fraction biological population model through contrivance by a couple of important and magnificent techniques, namely, the modified extended tanh-function method which depend on the balance theory and the Ricatti–Bernoulli sub-ODE method which is independent of the balancing principle. The suggested model is one of the major concerns for studying population distribution dynamics as well as the quantum field theory which is an important discipline for the description of interactions between light and electrons. The two suggested reliable, effective techniques are considered famous among ths ansatz methods that have various visions to realize the exact solutions to the non linear partial differential equation that reduce the volume of calculations examined before and usually give good results. It is solicited for this contrivance finding new exact solutions for two models in terms of some variable. The models are significant in quantum field theory, description of interactions between light and electron, quantum electrodynamics, demographic model, important to bring it into line with the reasonable distribution of wealth, resources, income, etc. The achieved results predict many types of solutions as trigonometric functions, hyperbolic functions, perfect periodic soliton solutions, singular periodic soliton solutions, and other rational solitons solutions. The efficiency of the techniques is demonstrated by the satisfactory results obtained through the derivation of closed-form soliton solutions from the exact solution by assigning definite values to the variables present in it.

In this article, we employ the Black Scholes model which plays a vital role in economic operation and financial market management. The Paul-Painlevé approach is used for the first time to achieve the exact wave solution to this equation. Furthermore, the numerical solution to this equation has been constructed by using the variational iteration method. [ABSTRACT FROM AUTHOR]

In this article, we employ the Mikhailov–Novikov–Wang integrable equation (MNWIE) appearing by means of the perturbatives symmetry approach to the rating of integrable non‐evolutionary PDEs. The new exact soliton solutions of this equation which were not achieved before have been realized for the first time in the framework of the (G′/G)‐expansion method. In the same vein and parallel, the corresponding numerical solutions of this equation have been established using the variational iteration method (VIM). [ABSTRACT FROM AUTHOR]

In this work, new varieties of unexpected scenarios for the solitons arising from the spatio-temporal dispersion (1 + 1)-dimensional Ito-equation (STDIE) are considered as an extension of the KdV (mKdV) type to higher order and usually employed to predict the rolling behavior of ships in regular sea. Moreover, they can describe the interaction process of two internal long waves specially the height of the water's free surface above a flat bottom where Υ (x , t) is an analytic function. We will introduce these new unexpected and impressive scenarios for the solitons that emerged from this model for the first time in the framework of three distinct schemas. The three schemas that are employed for this target are the Paul-Painleve approach method (PPAM), the extended direct algebraic method (EDAM) and ( G ′ /G)-expansion method. Our obtained solitons are new compared with that achieved before by other authors who applied various schemas. [ABSTRACT FROM AUTHOR]

In this study, we will derive many new diverse performances for the solitary wave solutions to the DNA Peyrard–Bishop Model with Beta-Derivative (DNAPBM) via three distinctive techniques. The first one has profile name: The ( G ′ / G) -expansion method, while the second one has the profile name: the extended direct algebraic method (EDAM) and the third one has the profile name: the Paul-Painlevé approach method (PPAM). The three techniques are implemented in the same vein and are parallel. The agreement between the achieved solitary wave solutions by these suggested three techniques has been shown. The novelty of our results will appear when the comparison is implemented between our obtained solutions and that achieved previously by other authors who applied another technique. [ABSTRACT FROM AUTHOR]

In this work, we will design unexpected configurations for the optical soliton propagation in lossy fiber system in the presence of the dispersion term solitons via two distinct and impressive techniques. The first one is the (G′/G)‐expansion method, while the second is solitary wave ansatze method. The two methods are implemented in the same vein and parallel. The obtained perceptions are new and were not achieved before. The comparison between our achieved visions and that achieved by other authors who used different schemas has been documented. [ABSTRACT FROM AUTHOR]

In this article, new variety types of exact solution to the Fujimoto- Watanable- equation (FWE) that equivalent to the modified Korteweg- de Vries- equation have been derived. These new types of solutions which weren't realized before by any other technique have been established in the framework of the Ricatti-Bernolli Sub-ODE method (RBSODM). Also, the identical numerical solutions whose initial conditions are emerged from the achieved exact solutions have been constructed by using the famous numerical variational iteration method (VIM). [ABSTRACT FROM AUTHOR]

The propagation of solitons in birefringent fiber is one of the phenomena that has an important role in all modern technological means of communication. The generalized (2 + 1) nonlinear Schrödinger equation (GNLSE) with its four-mixing waves (FMW) is the famous model that describes the propagation of solitons in birefringent fiber perfectly. In fact, the FMW governed effectively the performance of the resultant solitons amplitude. Hereby, we will study this model to extract new unexpected optical soliton solutions to this model via various three techniques. The three famous methods that are a candidate for this purpose are the extended direct algebraic method (EDAM), the extended simple equation method (ESEM) and the solitary wave ansatz method (SWAM). The three techniques are implemented successively for the suggested model to establish the optical solutions of the suggested model successfully. The optical soliton solutions that are achieved by these proposed techniques give surprise expectations that weren't achieved previously by any other authors who used other techniques. [ABSTRACT FROM AUTHOR]

In this article, we will implement the(G'=G)-expansion method which is used for the first time to obtain new optical soliton solutions of the thin-film ferroelectric materials equation (TFFME). Also, the numerical solutions of the suggested equation according to the variational iteration method (VIM) are demonstrated effectively. A comparison between the achieved exact and numerical solutions has been established successfully. [ABSTRACT FROM AUTHOR]