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We will study the transmission of double-hump solitons of the coupled Manakov equations (CME) that have unlimited position for the generation and amplification of double-hump solitons in fiber lasers. These double-hump soliton solutions of CME will be extracted by using three ansatz techniques that are prepared for this target. The first one has profile name the solitary wave ansatz method (SWAM), while the second one has profile name the extended simple equation method (ESEM) and the third one has profile name the Paul-Painlevé approach method (PPAM). The three proposed methods are used for the first time to construct the soliton solutions of this model. The simulation for the 2D, 3D Figures for all obtained solutions has been established. [ABSTRACT FROM AUTHOR]

Under the consideration of computational software Mathematica on the basis of extended simple equation method, constructed the periodic wave soliton solutions for nonlinear Zakharov–Kuznetsov modified equal width equation. In this study, constructed soliton solutions included periodic singular solitons, bright solitons, kink solitons, anti-kink solitons, travelling waves, dark solitons and solitary waves are more interesting, prices and some are novel. The graphical demonstration of some investigated results shown by contour, 2-dim and 3-dim by computing software Mathematica under numerical simulation to visualizing the physical phenomena of obtained results. The demonstrated results will be helpful in the study of nonlinear phenomena in various kinds of domains including nonlinear optics, nonlinear dynamics, solitons wave theory, optical fiber, nonlinear acoustic wave, communications system, networking system and transmission of signals. The extended simple equation technique is a concise, straightforward, powerful and reliable approach for the investigation of different kinds of nonlinear partial evolution equations. [ABSTRACT FROM AUTHOR]

In this article, new traveling wave solutions are retrieved for two space-time fractional models: the resonant nonlinear Schrödinger equation and the Biswas–Milovic equation which illustrates the dynamics of optical soliton promulgation in optical fibers, both featuring Kerr law nonlinearity. These equations are explored via an efficient method namely, the extended simple equation method. The fractional derivative is used in the conformable sense to accomplish this analysis. The extracted solutions show dark, periodic, singular, and singular-periodic solitons behaviors, which are depicted graphically by using line, surface, and contour plots. The reported solutions are unique and novel. The proposed method distinguishes itself by its simplicity, reliability, and ability to generate novel soliton solutions for nonlinear Schrödinger equations within the realm of mathematical physics. [ABSTRACT FROM AUTHOR]

In this research paper, explored the newly exact soliton solutions of the nonlinear fractional Kairat-II equation under extended simple equation method. The set of Kairat-II equations recently introduced, which have important applications in the area of physics and engineering. The fractional Kairat-II equation is an integrable equation and it is used to explain the differential geometry of curves and equivalence aspects. In this study, varieties of exact soliton solutions explored by apply the extension form of simple equation method on base of computational simulation. The extracted soliton solutions to the nonlinear fractional Kairat-II equation having interested physical structure in periodic optical solitons, bright solitons, kink solitons, anti-kink solitons, traveling waves, dark solitons, mixed solitons, peakon solitons and solitary waves. The physical structure of extracted soliton solutions visualized in 2-dim, 3-dim and contour graphically with the help of numerical simulation under computational software Mathematica. The investigation show that the proposed approach is more effective, powerful, concise and efficient for the other nonlinear partial and fractional differential equations. [ABSTRACT FROM AUTHOR]

This paper explore the new solitary wave solutions of the Biswas–Milovic equation and resonant nonlinear Schr o ¨ dinger equations with Kerr–Law nonlinearity. In mathematical physics, the Biswas–Milovic equation plays an important role. The dynamics of these solitons are well known to propagate over large distances in a few femto-seconds through these fibres. These dynamics are governed by the nonlinear Schrödinger's equation. The resonant nonlinear Schrödinger equation can be used to model the propagation of waves in fiber optics. The powerful analytical approach known as the extended simple equation method is employed to explore the solitary wave solutions of the Biswas–Milovic and resonant nonlinear Schr o ¨ dinger equations. These equations are primarily explored in the realm of solitons in nonlinear fiber optics. New dark, kink, anti-kink and singular periodic solitons are secured. Furthermore, 3D surface graphs, contour plots and parametric graphs are drawn to show dynamical behavior of some obtained solutions with the aid of symbolic software such as Mathematica. The constructed solutions will helpful to understand the dynamical framework of nonlinear Schr o ¨ dinger equations in the related physical phenomena. [ABSTRACT FROM AUTHOR]

The main purpose of this study was to produce abundant new types of soliton solutions for the Radhakrishnan-Kundu-Lakshmanan equation that represents unstable optical solitons that emerge from optical propagations through the use of birefringent fibers. These new types of soliton solutions have behaviors that are bright, dark, W-shaped, M-shaped, periodic trigonometric, and hyperbolic and were not realized before by any other method. These new forms have been detected by using four different techniques, which are, the extended simple equation method, the Paul-Painlevé approach method, the Ricatti-Bernoulli-sub ODE, and the solitary wave ansatz method. These new solitons will be arranged to create a soliton catalog with new impressive behaviors and they will contribute to future studies not only for this model but also for the optical propagations through birefringent fiber.

In our current article, we will use two diverse methods namely the extended simple equation method (ESEM) and the extended direct algebraic method (EDAM) to extract the soliton solutions of popularized anti-cubic nonlinear Schrödinger equation that is very useful in the field of the optics. The obtained rational solutions via these two reliable, effective techniques denote the importance of these methods. Moreover, we will implement the differential transform method (DTM) which is one of the most, new semi-analytical and numerical methods to construct the corresponding numerical solutions for all achieved soliton solutions by the above two methods. We will compare between the soliton solutions introduced by the two suggested methods with the numerical solutions obtained by the DTM. It is clear that there exist similarity and convergence between the traveling wave solutions achieved by the ESEM, EDAM and the numerical solutions achieved by DTM. The novelty of our achieved solutions will appear when it compared by [1]. [ABSTRACT FROM AUTHOR]

The main purpose of this study was to produce abundant new types of soliton solutions for the Radhakrishnan-Kundu-Lakshmanan equation that represents unstable optical solitons that emerge from optical propagations through the use of birefringent fibers. These new types of soliton solutions have behaviors that are bright, dark, W-shaped, M-shaped, periodic trigonometric, and hyperbolic and were not realized before by any other method. These new forms have been detected by using four different techniques, which are, the extended simple equation method, the Paul-Painlevé approach method, the Ricatti-Bernoulli-sub ODE, and the solitary wave ansatz method. These new solitons will be arranged to create a soliton catalog with new impressive behaviors and they will contribute to future studies not only for this model but also for the optical propagations through birefringent fiber. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some new traveling wave solutions to Vakhnenko-Parkes equation via three modified mathematical methods. The derived solutions have been obtained including periodic and solitons solutions in the form of trigonometric, hyperbolic, and rational function solutions. The graphical representations of some solutions by assigning particular values to the parameters under prescribed conditions in each solutions and comparing of solutions with those gained by other authors indicate that these employed techniques are more effective, efficient and applicable mathematical tools for solving nonlinear problems in applied science. [ABSTRACT FROM AUTHOR]

In this work, we construct traveling wave solutions of (1+1) - dimensional Ito integro-differential equation via three analytical modified mathematical methods. We have also compared our achieved results with other different articles. Portrayed of some 2D and 3D figures via Mathematica software demonstrates to understand the physical phenomena of the nonlinear wave model. These methods are powerful mathematical tools for obtaining exact solutions of nonlinear evolution equations and can be also applied to non-integrable equations as well as integrable ones. Hence worked-out results ascertained suggested that employed techniques best to deal NLEEs.

In this survey, the ionic currents along the microtubules equation handled by applying two different techniques. It describes the ionic transport throughout the intracellular environment. This phenomenon explains the behavior of many applications in a biological nonlinear dispatch line for ionic currents. Using two different analytical techniques on this equation allows us to get new forms of analytical solitary traveling wave solutions. The obtained solutions support many researchers concerned with the discussion of the physical properties of the ionic currents along microtubules. Microtubules are one of the main components of the cytoskeleton, and function in many operations, comprehensive constitutional backing, intracellular transmit, and DNA division. The thing which makes us applies this property in many applications for this model. The earned results demonstrate that the proposed methods are significantly powerful, evangelist, relaxing, suitable, and convenient for solving many nonlinear models. [ABSTRACT FROM AUTHOR]

In this article, the extended simple equation method is employed to construct solitary wave solutions of Boussineq equation, which describes gravity waves propagation on the surface of water and also explains the collision of oblique waves transformation. The extended simple equation method is a new technique which helpful to other sorts of nonlinear evolution equations in current areas of research in mathematics and physics. [ABSTRACT FROM AUTHOR]

In this work, we apply the extended simple equation method to study the dispersive traveling wave solutions of (2+1)-dimensional Nizhnik–Novikov–Vesselov (NNV), Caudrey–Dodd–Gibbon (CDG) and Jaulent–Miodek (JM) hierarchy nonlinear equations. A set of exact, periodic and soliton solutions is obtained for these models confirming the effectiveness of the proposed method. The models studied are important for a number of application areas especially in the field of mathematical physics. Interesting figures are used to illustrate the physical properties of some obtained results. A comparison between obtained solutions and established results in the literature is also given. [ABSTRACT FROM AUTHOR]

The extended simple equation method is applied to construct solitary wave solutions of (3 + 1)-dimensional Kadomtsev–Petviashvili-Benjamin–Bona–Mahony (KP-BBM), Korteweg–de Vries Benjamin–Bona–Mahony (KdV-BBM), Breaking soliton (BS) and (2 + 1) Maccari system waves system of equations. These models have prevalent usage in modern science. This technique can also be functional to solve different kinds of nonlinear evolution problems in contemporary areas of research. It is an effective and powerful mathematical tool in finding solitary wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics. [ABSTRACT FROM AUTHOR]

In this research, we study the generalized Radhakrishnan–Kundu–Lakshmanan equation which describe the dynamics of light pulses and it is also described by the nonlinear Schrödinger family of equations with cubic nonlinear terms. We use two auxiliary equation methods (extended simple equation method and new auxiliary equation method)) to obtain the exact and solitary traveling wave solutions of this model. We get new form of solitary traveling wave solutions for this model which help the researchers who interested in the applications of this model to be able to study the physical properties of the model. [ABSTRACT FROM AUTHOR]

In this research, we study new two techniques that called the extended simple equation method and the novel G ′ G -expansion method. The extended simple equation method depend on the auxiliary equation d ϕ d ξ = α + λ ϕ + μ ϕ 2 which has three ways for solving depends on the specific condition on the parameters as follow: When λ = 0 this auxiliary equation reduces to Riccati equation, when α = 0 this auxiliary equation reduces to Bernoulli equation and when α ≠ 0 , λ ≠ 0 , μ ≠ 0 we the general solutions of this auxiliary equation while the novel G ′ G -expansion method depends also on similar auxiliary equation G ′ G ′ = μ + λ G ′ G + ( v - 1 ) G ′ G 2 which depend also on the value of ( λ 2 - 4 ( v - 1 ) μ ) and the specific condition on the parameters as follow: When λ = 0 this auxiliary equation reduces to Riccati equation, when μ = 0 this auxiliary equation reduces to Bernoulli equation and when ( λ 2 ≠ 4 ( v - 1 ) μ ) we the general solutions of this auxiliary equation. This show how both of these auxiliary equation are special cases of Riccati equation. We apply these methods on two dimensional nonlinear Kadomtsev-Petviashvili Burgers equation in quantum plasma and three-dimensional nonlinear modified Zakharov-Kuznetsov equation of ion-acoustic waves in a magnetized plasma. We obtain the exact traveling wave solutions of these important models and under special condition on the parameters, we get solitary traveling wave solutions. All calculations in this study have been established and verified back with the aid of the Maple package program. The executed method is powerful, effective and straightforward for solving nonlinear partial differential equations to obtain more and new solutions. [ABSTRACT FROM AUTHOR]

The Equal-Width and Modified Equal-Width equations are used as a model in partial differential equations for the simulation of one-dimensional wave transmission in nonlinear media with dispersion processes. In this article we have employed extend simple equation method and the exp(−varphi( ξ )) expansion method to construct the exact traveling wave solutions of equal width and modified equal width equations. The obtained results are novel and have numerous applications in current areas of research in mathematical physics. It is exposed that our method, with the help of symbolic computation, provides a effective and powerful mathematical tool for solving different kind nonlinear wave problems. [ABSTRACT FROM AUTHOR]

In this current work, we employ novel methods to find the exact travelling wave solutions of Modified Liouville equation and the Symmetric Regularized Long Wave equation, which are called extended simple equation and exp(-Ψ( ξ ))-expansion methods. By assigning the different values to the parameters, different types of the solitary wave solutions are derived from the exact traveling wave solutions, which shows the efficiency and precision of our methods. Some solutions have been represented by graphical. The obtained results have several applications in physical science. [ABSTRACT FROM AUTHOR]

In this research, we investigate one of the most popular model in nature and also industrial which is the pressure equation of bubbly liquids with examination for viscosity and heat transfer which has many application in nature and engineering. Understanding the physical meaning of exact and solitary traveling wave solutions for this equation gives the researchers in this field a great clear vision of the pressure waves in a mixture liquid and gas bubbles taking into consideration the viscosity of liquid and the heat transfer and also dynamics of contrast agents in the blood flow at ultrasonic researches. To achieve our goal, we apply three different methods which are extended tanh-function method, extended simple equation method and a new auxiliary equation method on this equation. We obtained exact and solitary traveling wave solutions and we also discuss the similarity and difference between these three method and make a comparison between results that we obtained with another results that obtained with the different researchers using different methods. All of these results and discussion explained the fact that our new auxiliary equation method is considered to be the most general, powerful and the most result-oriented. These kinds of solutions and discussion allow for the understanding of the phenomenon and its intrinsic properties as well as the ease of way of application and its applicability to other phenomena. [ABSTRACT FROM AUTHOR]

This paper studies dual forms of integrability to reveal a spectrum of optical soliton solutions of a governing model. These are modified simple equation method and extended simple equation scheme. The model also incorporates perturbation terms that are of Hamiltonian type. Four forms of nonlinear refractive index structures are studied. [ABSTRACT FROM AUTHOR]

• Applications of mathematical physics models. • The hydrodynamic mathematical methods • First integro-differential KP hierarchy equation. In our research, we ascertain abundant novel exact traveling wave solutions of (2 + 1)-dimensional first integro-differential Kadomtsev-Petviashivili hierarchy equation by two new modified mathematical methods namely called generalized direct algebraic and extended simple equation methods with the help of computer package like Mathematica. These exact solutions are expressed in the forms of trigonometric, hyperbolic and rational function solutions which are established by using our eminent methods and also comparing our results to all of the well-known results that are given in the literature. These methods are more easier and having fruitful applications to solve different nonlinear wave problems in nonlinear science. [ABSTRACT FROM AUTHOR]