Hyperbolic Function Solutions Research Articles

Exact Solutions to Space-Time Fractional Fokas-Lenells Equations With Parameters

The exact solutions of the space-time fractional Fokas-Lenells equation with parameters in nonlinear optics are obtained by using the complete discrimination system for polynomial method, including rational function solutions, periodic solutions, solitary wave solutions, Jacobi elliptic function solutions and hyperbolic function solutions. The relevant images of the exact solution are drawn, and the influence of parameters on the structure of the solution is analyzed.

Novel Analytical Technique to Find Closed Form Solutions of Time Fractional Partial Differential Equations

In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential equation into an integer order ordinary differential equation. Using the novel (G′/G2)-expansion method, several exact solutions with extra free parameters are found in the form of hyperbolic, trigonometric, and rational function solutions. When parameters are given appropriate values along with distinct values of fractional order α travelling wave solutions such as singular periodic waves, singular kink wave soliton solutions are formed which are forms of soliton solutions. Also, the solutions obtained by the proposed method depend on the value of the arbitrary parameters H. Previous results are re-derived when parameters are given special values. Furthermore, for numerical presentations in the form of 3D and 2D graphics, the commercial software Mathematica 10 is incorporated. The method is accurately depicted, and it provides extra general exact solutions.

Hyperbolic function solutions of time-fractional Kadomtsev-Petviashvili equation with variable-coefficients

<abstract><p>Based on the variable separation method, the Kadomtsev-Petviashvili equation is transformed into a system of equations, in which one is a fractional ordinary differential equation with respect to time variable $ t $, and the other is an integer order variable coefficients partial differential equation with respect to spatial variables $ x, y $. Assuming that the coefficients of the obtained partial differential equation satisfy certain conditions, the equation is further reduced. The extended homogeneous balance method is used to find the exact solutions of the reduced equation. According to the solutions of some special fractional ordinary differential equations, we obtain some hyperbolic function solutions of time-fractional Kadomtsev-Petviashvili equation with variable coefficients.</p></abstract>

New Exact Traveling Wave Solutions of (2 + 1)-Dimensional Time-Fractional Zoomeron Equation

In this paper, the new mapping approach and the new extended auxiliary equation approach were used to investigate the exact traveling wave solutions of (2 + 1)-dimensional time-fractional Zoomeron equation with the conformable fractional derivative. As a result, the singular soliton solutions, kink and anti-kink soliton solutions, periodic function soliton solutions, Jacobi elliptic function solutions and hyperbolic function solutions of (2 + 1)-dimensional time-fractional Zoomeron equation were obtained. Finally, the 3D and 2D graphs of some solutions were drawn by setting the suitable values of parameters with Maple, and analyze the dynamic behaviors of the solutions.

Soliton solutions of conformable time-fractional perturbed Radhakrishnan-Kundu-Lakshmanan equation

<abstract><p>In this paper, our main purpose is to study the soliton solutions of conformable time-fractional perturbed Radhakrishnan-Kundu-Lakshmanan equation. New soliton solutions have been obtained by the extended $ (G'/G) $-expansion method, first integral method and complete discrimination system for the polynomial method, respectively. The solutions we obtained mainly include hyperbolic function solutions, solitary wave solutions, Jacobi elliptic function solutions, trigonometric function solutions and rational function solutions. Moreover, we draw its three-dimensional graph.</p></abstract>

A novel integration approach to study the perturbed Biswas-Milovic equation with Kudryashov’s law of refractive index

Modelling the movement of waves and the propagation of waves is extremely important in the field of fluid dynamics. All types of waves phenomena are very important in real-life implementations. We use an efficient solver and the improved F-expansion approach to obtain precise traveling wave solutions to the perturbed Biswas-Milovic equation with Kudryashov's law of refractive index. All acquired solutions are in the form of trigonometric, hyperbolic, and rational function solutions, and the resulting solutions are applicable to real-world problems in fluid dynamics, optical fibers, plasma physics, and so on. Furthermore, we depicted the three-dimensional and contour graphs to illustrate the behavior of some obtained solutions.

Dynamics of optical pulses in fiber optics

In this paper, we pay attention to the nonlinear dynamical behavior of ultra-short pulses in optical fiber. The new Hamiltonian amplitude equation is used as a governing model to analyze the pulses. We secure the ultra-short pulses in the forms of bright, dark, singular, combo and complex soliton solutions by the utilization of three of sound computational integration techniques that are protracted (or extended) Fan-sub equation method (PFSEM), the generalized exponential rational function method (GERFM), extended Sinh-Gordon equation expansion method (ShGEEM). Moreover, Jacobi’s elliptic, trigonometric, and hyperbolic functions solutions are also discussed as well as the constraint conditions of the achieved solutions are also presented. In addition, we discuss the different wave structures by the assistance of logarithmic transformation. The findings demonstrate that the examined equation theoretically contains a large number of soliton solution structures. By selecting appropriate criteria, the actual portrayal of certain obtained results is sorted out graphically in 3D and 2D profiles. The results suggest that the procedures used are concise, direct, and efficient, and that they can be applied to more complex phenomena. The resulting solutions are novel, intriguing, and potentially useful in understanding energy transit and diffusion processes in mathematical models of several disciplines of interest, including nonlinear optics. Our new results have been compared to these in the literature.

Classification of All Single Traveling Wave Solutions of Fractional Coupled Boussinesq Equations via the Complete Discrimination System Method

In this paper, the complete discrimination system method is used to construct the exact traveling wave solutions for fractional coupled Boussinesq equations in the sense of conformable fractional derivatives. As a result, we get the exact traveling wave solutions of fractional coupled Boussinesq equations, which include rational function solutions, Jacobian elliptic function solutions, implicit solutions, hyperbolic function solutions, and trigonometric function solutions. Finally, the obtained solution is compared with the existing literature.

Deeper properties of the nonlinear Phi-four and Gross-Pitaevskii equations arising mathematical physics

In this paper, the rational sine–cosine and rational sinh–cosh methods are applied in extracting some properties of nonlinear Phi-four and Gross–Pitaevskii equations. The singular periodic wave solutions, dark soliton solutions and hyperbolic function solutions are reported. The solitary waves are observed from the traveling waves under the values of the parameters. Modulation instability analysis is also observed in various simulations. We also plot to observe the wave distributions of parameters of stability in 2D and 3D visuals via package program.

Specific wave profiles and closed-form soliton solutions for generalized nonlinear wave equation in (3+1)-dimensions with gas bubbles in hydrodynamics and fluids

Nonlinear evolution equations (NLEEs) are frequently employed to determine the fundamental principles of natural phenomena. Nonlinear equations are studied extensively in nonlinear sciences, ocean physics, fluid dynamics, plasma physics, scientific applications, and marine engineering. The generalized exponential rational function (GERF) technique is used in this article to seek several closed-form wave solutions and the evolving dynamics of different wave profiles to the generalized nonlinear wave equation in (3+1) dimensions, which explains several more nonlinear phenomena in liquids, including gas bubbles. A large number of closed-form wave solutions are generated, including trigonometric function solutions, hyperbolic trigonometric function solutions, and exponential rational functional solutions. In the dynamics of distinct solitary waves, a variety of soliton solutions are obtained, including single soliton, multi-wave structure soliton, kink-type soliton, combo singular soliton, and singularity-form wave profiles. These determined solutions have never previously been published. The dynamical wave structures of some analytical solutions are graphically demonstrated using three-dimensional graphics by providing suitable values to free parameters. This technique can also be used to obtain the soliton solutions of other well-known equations in engineering physics, fluid dynamics, and other fields of nonlinear sciences.

Investigation of lump, soliton, periodic, kink, and rogue waves to the time-fractional phi-four and (2+1) dimensional CBS equations in mathematical physics

In this article, we investigate the lump, soliton, periodic, kink, and rogue waves to the time-fractional phi-four and (2+1) dimensional Calogero-Bogoyavlanskil schilf (CBS) equations. The (G′/G,1/G)-expansion technique is applied to obtain the solutions of the phi-four and Calogero-Bogoyavlanskil schilf (CBS) equations with the contribution of conformable derivative. The stated method declines a huge volume of calculation. The (G′/G,1/G)-expansion technique is also known as the two-variable method. Consequently, lump, mixed lump, soliton, bell-shaped soliton, periodic, kink, periodic-kink, singular kink, rogue wave solutions are displayed in hyperbolic and trigonometric function solutions. Varieties of other traveling wave solutions are also formulated with the mentioned method. Moreover, for the physical explanation of the obtained solutions, the three-dimensional (3D) surface plots, contour plots combined with 3D surface plots, and two-dimensional (2D) plots are presented using the computer software MATLAB. Additionally, it is possible to deduce that the obtained solutions and their physical properties will aid in the understanding of water wave circulation in nonlinear dynamics.

Newly Developed Analytical Scheme and Its Applications to the Some Nonlinear Partial Differential Equations with the Conformable Derivative

This paper presents a novel and general analytical approach: the rational sine-Gordon expansion method and its applications to the nonlinear Gardner and (3+1)-dimensional mKdV-ZK equations including a conformable operator. Some trigonometric, periodic, hyperbolic and rational function solutions are extracted. Physical meanings of these solutions are also presented. After choosing suitable values of the parameters in the results, some simulations are plotted. Strain conditions for valid solutions are also reported in detail.

Periodic wave analysis to the time-fractional phi-four and (2+1) dimensional CBS equations

In this paper, we investigate some innovative and exact travelling wave solutions to the time-fractional phi-four equation and the (2+1) dimensional Calogero-Bogoyavlanskil schilf (CBS) equation using the ( )-expansion method. Periodic solutions are displayed in hyperbolic, trigonometric, and rational function solutions. Subsequently, we construct some new solutions connecting the free parameters of the phi-four equation and the CBS equation, which are characterized into three complete forms: rational function, trigonometric function, and hyperbolic functions. Graphical representations of some attained solutions are also presented in this article. Hence, this study shows the efficiency and the easiness of the ( )-expansion technique with the assistance of emblematically computational software MATLAB and Mathematica. Â

Evolutionary dynamics of solitary wave profiles and abundant analytical solutions to a (3+1)-dimensional burgers system in ocean physics and hydrodynamics

In the fields of oceanography, hydrodynamics, and marine engineering, many mathematicians and physicists are interested in Burgers-type equations to show the different dynamics of nonlinear wave phenomena, one of which is a (3+1)-dimensional Burgers system that is currently being studied. In this paper, we apply two different analytical methods, namely the generalized Kudryashov (GK) method, and the generalized exponential rational function method, to derive abundant novel analytic exact solitary wave solutions, including multi-wave solitons, multi-wave peakon solitons, kink-wave profiles, stripe solitons, wave-wave interaction profiles, and periodic oscillating wave profiles for a (3+1)-dimensional Burgers system with the assistance of symbolic computation. By employing the generalized Kudryashov method, we obtain some new families of exact solitary wave solutions for the Burgers system. Further, we applied the generalized exponential rational function method to obtain a large number of soliton solutions in the forms of trigonometric and hyperbolic function solutions, exponential rational function solutions, periodic breather-wave soliton solutions, dark and bright solitons, singular periodic oscillating wave soliton solutions, and complex multi-wave solutions under various family cases. Based on soft computing via Wolfram Mathematica, all the newly established solutions are verified by back substituting them into the considered Burgers system. Eventually, the dynamical behaviors of some established results are exhibited graphically through three - and two-dimensional wave profiles via numerical simulation.

Closed form solutions of nonlinear space‐time fractional Drinfel'd‐Sokolov‐Wilson equation via reliable methods

In current study, the closed form solutions for nonlinear space‐time fractional Drinfel'd‐Sokolov‐Wilson (FDSW) equation are obtained in the presence of the beta derivatives and using reliable methods. Applying the (G′/G)‐expansion method, the hyperbolic, trigonometric, and rational function solutions of space‐time FDSW equation are derived. It is interesting to propose the generalized complex method to construct the closed form solutions of nonlinear fractional differential equations (NFDEs), and it is a good example to apply for searching exact solutions of space‐time FDSW equation. Computer simulations of solutions obtained via these two reliable methods are also presented by some figures with different values of the parameter beta. The results reveal that these two methods are efficient and direct approaches for solving various fractional differential equations in mathematical physics.

Protracted study on a real physical phenomenon generated by media inhomogeneities

In this work, we study the dynamical behavior for a real physical application due to the inhomogeneities of media via analytical and numerical approaches. This phenomenon is described by the 3D Date–Jimbo–Kashiwara–Miwa (3D-DJKM) equation. For analytical techniques, three different methods are performed to get hyperbolic, trigonometric and rational functions solutions. After that, the obtained solutions are graphically depicted through 2D- and 3D-plots and numerically compared via the finite difference algorithm to check the precision of the proposed methods.

Dynamical behaviors of various optical soliton solutions for the Fokas–Lenells equation

In this work, the new optical soliton solutions and interaction solutions for the space-time fractional Fokas–Lenells equation with fractional [Formula: see text]-derivatives are constructed via three mathematical analytical techniques, namely the extended SE method, unified solver method, and three-wave methods. The results have proved the efficiency of the suggested techniques for obtaining abundant optical soliton solutions to nonlinear evolution equations (NLEEs) and closed-form solutions in the forms of rational function solutions; hyperbolic and trigonometric function solutions and multi-wave interaction solutions are obtained. These techniques are more efficient, robust, and powerful mathematical tools for acquiring several optical soliton solutions for many other fractional space-time NLEEs that arise in optical physics and plasma physics. The graphical representations of the combined optical solitons are demonstrated using three- and two-dimensional graphics.

Soliton solutions, travelling wave solutions and conserved quantities for a three-dimensional soliton equation in plasma physics

Many physical systems can be successfully modelled using equations that admit the soliton solutions. In addition, equations with soliton solutions have a significant mathematical structure. In this paper, we study and analyze a three-dimensional soliton equation, which has applications in plasma physics and other nonlinear sciences such as fluid mechanics, atomic physics, biophysics, nonlinear optics, classical and quantum fields theories. Indeed, solitons and solitary waves have been observed in numerous situations and often dominate long-time behaviour. We perform symmetry reductions of the equation via the use of Lie group theory and then obtain analytic solutions through this technique for the very first time. Direct integration of the resulting ordinary differential equation is done which gives new analytic travelling wave solutions that consist of rational function, elliptic functions, elementary trigonometric and hyperbolic functions solutions of the equation. Besides, various solitonic solutions are secured with the use of a polynomial complete discriminant system and elementary integral technique. These solutions comprise dark soliton, doubly-periodic soliton, trigonometric soliton, explosive/blowup and singular solitons. We further exhibit the dynamics of the solutions with pictorial representations and discuss them. In conclusion, we contemplate conserved quantities for the equation under study via the standard multiplier approach in conjunction with the homotopy integral formula. We state here categorically and emphatically that all results found in this study as far as we know have not been earlier obtained and so are new.

Abundant analytical soliton solutions and Evolutionary behaviors of various wave profiles to the Chaffee–Infante equation with gas diffusion in a homogeneous medium

The generalized exponential rational function method is used in this work to obtain a variety of rich families of analytical soliton solutions and to exemplify the dynamics of solitonic structures of the (2+1)-dimensional ChaffeeInfante (CI) equation. This CI equation is a well-known reaction–diffusion equation that can describe the physical process of mass transport and particle diffusion and has been widely used in fluid dynamics, electromagnetic wave fields, high-energy physics, fluid mechanics, coastal engineering, and ion-acoustic waves in plasma physics, optical fibers, and other disciplines. The GERF method is successfully used to derive closed-form analytic solutions to the CI equation, including rational, exponential, trigonometric, and hyperbolic function solutions. The CI equation has indeed been observed to have bright and dark solitons, singular and combined singular soliton profiles, periodic oscillating nonlinear waves, and kink-wave profiles. Furthermore, three-dimensional graphical representations are presented to depict the dynamical behavior of the achieved solutions. We can better understand the dynamical properties and structures of these solutions by using these three-dimensional postures. This technique could be used to solve a wide variety of higher-dimensional nonlinear equations confronted in mathematical physics and applied sciences.

On some novel solution solutions to the generalized Schrödinger-Boussinesq equations for the interaction between complex short wave and real long wave envelope

This paper explores some novel solutions to the generalized Schrödinger-Boussinesq (gSBq) equations, which describe the interaction between complex short wave and real long wave envelope. In order to derive some novel complex hyperbolic and complex trigonometric function solutions, the sine-Gordon equation method (sGEM) is applied to the gSBq equations. Novel complex hyperbolic and trigonometric function solutions are expressed by the dark, bright, combo dark-bright, W-shaped, M-shaped, singular, combo singular, and periodic wave solutions. The accuracy of the explored solitons is examined under the back substitution to the corresponding equations via the symbolic computation software Maple. It is found from the back substitution outcomes that all soliton solutions satisfy the original equations. The proper significance of the explored outcomes is demonstrated by the three-dimensional (3D) and two-dimensional (2D) graphs, which are presented under the selection of particular values of the free parameters. All the combo-soliton (W-shaped, M-shaped, and periodic wave) solutions are found to be new for the interaction between complex short wave and real long wave envelope in laser physics that show the novelty of the study. Moreover, the applied method provides an efficient tool for exploring novel soliton solutions, and it overcomes the complexities of the solitary wave ansatz method.