From the point of view of three famous and important techniques we will achieve new expectations for the soliton configurations to the (2+1)-dimensional generalized nonlinear Schrödinger equation (GNLSE) with four waves mixing (FWM). The suggested model describes propagation of solitons in birefringent fiber. The FWM governed effectively the performance of the resultant soliton amplitudes. The three famous methods candidates 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 successfully. Surprise expectations for solitons via these three techniques to this model which weren’t achieved previously by any other authors who used other techniques have been demonstrated.

The Nagumo equation describes a reaction-diffusion system in biology. Here, it is coupled to Burgers equation, via including convection, which is, namely; Burgers-Nagumo equation BNE. The first objective of this work is to present a theorem to reduce the different versions of the fractional time derivatives FTD to “non autonomous” ordinary ones, that is ordinary derivatives with time dependent coefficients. The second objective is to find the exact solutions of the fractal and fractional time derivative -BNE, that is to solve BNE with time dependent coefficient. On the other hand FTD can be transformed to BNE with constant coefficients via similarity transformations. The unified and extended unified method are used. Self-similar solutions are also obtained. It is found that significant fractal effects hold for smaller order derivatives. While significant fractional effects hold for higher-order derivatives. The solutions obtained show solitary, wrinkle soliton waves, with double kinks, undulated, or with spikes. Further It is shown that wrinkle soliton wave, with double kink configuration holds for smaller fractal order. While in the case of fractional derivative, this holds for higher orders.

In this study, we will implement new perceptions for the bright and dark soliton solutions to the modified nonlinear Schrödinger equation (MNLSE)or forms of the rogue wave modes for a derivative nonlinear Schrodinger model with positive linear dispersion which describe the propagation of rogue waves in Ocean engineering as well as all similar waves such as dynamics waveguides that have unexpected large displacements, the waves which occur only in the regime of positive cubic nonlinearity, regime that coincides exactly with the existence of instabilities of plane waves , long-wave limit of a breather (a pulsing mode). Two famous different schemas are involved for this purpose. The first schema is the solitary wave ansatze method (SWAM), while the second scheme is the extended simple equation method (ESEM). The two schemas are implemented in the same vein and parallel to construct new perceptions to the soliton solutions of this model. A comparison between the obtained new perceptions with the old perceptions that achieved previously by other authors has been demonstrated.

Although fractional and classical order cubic quintic nonlinear Schrödinger (NS) equation and cubic nonlinear Schrödinger equation are used simultaneously in nonlinear optics disciplines, the fractional-order NS equations are nowadays extensively used due to their higher coherence. The space-time fractional cubic quintic and nonlinear cubic Schrödinger equations integrating beta derivative are significant in modeling to nonlinear optics, photonics, plasmas, condensed matter physics, and other domains. The fractional wave transformation is exploited to translate the space-time fractional equations and the optical soliton solutions in the form of exponential, trigonometric, and hyperbolic functions with free parameters have been established in this article by putting to use the improved Bernoulli sub-equation function (IBSEF) approach. The shape of the solutions includes kink, periodic, bell-shaped soliton, breathing soliton, bright soliton, and singular kink type soliton. The physical features of the solitons have been revealed by depicting 3D, 2D, contour, and density graphs of some of the solutions. The results demonstrate that the IBSEF approach is simple, straightforward, effective and that it can be applied to a wide range of nonlinear fractional-order models in optics and communication engineering to achieve soliton solutions.

In this work, we will design unexpected configurations for the optical soliton propagation in lossy fiber system in presence 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 same vein and parallel. The obtained perceptions are new and weren’t achieved before. The comparison between our achieved visions and that achieved by other authors who used different schemas has been documented.

This paper is about the investigation of exact solutions of important economic model; Ivancevic option pricing model (IOPM) with M-fractional derivative. To achieve this aim, three different methods; expa function method, extended Sinh-Gordon equation expansion method (EShGEEM) and extended (G′/G)-expansion method are used. Obtained solutions consisting of trigonometric, hyperbolic trigonometric, rational and exponential. The obtained solutions are new than the existing solutions in the literature. The got solutions are also verified by using Mathematica tool. Graphically justification are also done by plotting 2-D,3-D and contour graphs. The importance of this paper is that M-fractional derivative is first time use for this model. On the bases of achieved results it is suggested that these methods are simple, reliable and fruitful than the other methods.

This paper aims to uncover fairly interesting optical soliton solutions in (2+1)-dimensions. The fractional temporal Kundu-Mukherjee-Naskar (KMN) equation is reviewed as a governing model. Local M-derivative along with the unified approach is used to acquire these soliton solutions. The predicted solutions are yielded with the constraint conditions and highlighted by their graphical portrayal. Lastly, the inuence of a local fractional parameter upon predicted solutions are depicted through 2D and 3D graphs.

From point of view of two different schemas several new impressive lump solutions to (1+1)-dimensional Ito equation have been established. The first schema is the Paul-Painleve approach method (PPAM) which will be applied perfectly to extract multiple lump solutions of this model, while the second schema is the famous one of the ansatze method and has personal profile named the Ricatti-Bernolli Sub-ODE method. In related subject the numerical solutions corresponding to all lump solutions achieved via each method have been demonstrated individually in the framework of the variational iteration method (VIM).

In this article, the Paul-Painleve approach (PPA) which discovered recently and built on the balance role has been used perfectly to achieve new impressive solitary wave solutions to the nano-soliton of ionic waves (NSOIW) propagating along microtubules in living cells. In addition, the variational iteration method (VIM) has been applied in the same vein and parallel to establish the numerical solutions of this model.

In this article, imperssive exact solutions and hence effective regulations to the non-fractional order and the time-fractional order of the biological population models are achevied for the first time in the framwork of the Paul-Painlevé approach. When the variables appearing in the exact solutions take specific values, the solaitry wave solutions will be easily satisfied.The realized results prove the efficiency of this technique.

In this article, we employ the nonlinear complex Hirota-dynamical model which is one of the famous and important standards to the nonlinear Schrödinger equation in which the third derivative term represent the self-interaction in the high-frequency subsystem. Specially, in plasma this term is isomorphic to the so known self-focusing effect. The bright, dark and periodic optical soliton solutions to this equation will realized successfully for the first time in the framework of the solitary wave ansatz method. Furthermore, in this connection at the same time and parallel the extended simple equation method has been applied successfully to achieve new impressive solitary wave solutions to this model. A comparison between the obtained results and that satisfied in previous work has been established.

This paper is about the study of space-time fractional Fokas-Lenells equation that describes nonlinear wave propagation in optical fibers. Three prominent schemes are employed for extracting different types of exact soliton solutions. In particular, the expa function method, the hyperbolic function method and the simplest Riccati equation scheme are investigated for the said model. As a sequela, a series of soliton solutions are obtained and verified through Mathematica. The obtained solutions are significant additions in some specific fields of physics and engineering. Furthermore, the 3D graphical descriptions are left to analyze the pulse propagation for the reader.