Hyperbolic Function Solutions Research Articles

Bifurcation and new exact traveling wave solutions for time-space fractional Phi-4 equation

In this paper, the dynamical behavior of a time-space fractional Phi-4 equation is investigated by utilizing the bifurcation method of a planar dynamical system. Under the given parameter conditions, phase portraits and bifurcations are obtained with the help of the mathematical software Maple. Moreover, some new exact traveling wave solutions are obtained, such as Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions, kink solitary wave solutions, and periodic wave solutions.

Exact and explicit travelling-wave solutions to the family of new 3D fractional WBBM equations in mathematical physics

In this paper, the new 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations newly announced are investigated with the introduction of the spatial and sequential fractional-order derivatives expending in the sense of conformable derivative. Several solutions containing hyperbolic and trigonometric function solutions are created using the modified extended tanh-function (mETF) method. The productivity of the mETF process for building these solutions has been established. This method decreases the large volume of calculations. With the assist of computational software MALAB, the solutions' graphical exemplifications are illustrated in three-dimension (3D) and two-dimension (2D) plot. All plots represent the absolute wave configurations through the obtained solutions to the mentioned equation, preferring suitable parameters. Moreover, it may conclude that the attained solutions and their physical features might help comprehend the shallow water wave propagation in nonlinear dynamics.

Optical solitons of Biswas-Arshed equation in birefringent fibers using extended direct algebraic method

In this paper, we applied the extended direct algebraic method (EDAM) to examine the dark, singular, combined dark-bright, combined bright-singular and periodic singular solitons as well as hyperbolic and trigonometric functions solutions of Biswas-Arshed equation (BAE) in birefringent fibers. The outcomes reveal that our technique is very dynamic and straightforward. The determined solutions of this model are new in the literature.

Exact traveling wave solutions of Chaffee–Infante equation in (2 + 1)‐dimensions and dimensionless Zakharov equation

In this work, the generalized Kudryashov method is used to obtain the exact traveling wave solutions for two important nonlinear evolution equations, the Chaffee–Infante equation in (2 + 1)‐dimensions and the dimensionless Zakharov equation. The generalized Kudryashov method is successfully used for getting exact solutions of these nonlinear equations in the form of exponential function solutions and hyperbolic function solutions. Moreover, we have discussed the dynamical behaviors through graphical representation of the solutions obtained in this way.

On exact special solutions for the stochastic regularized long wave-Burgers equation

In this paper, we will analyze the Regularized Long Wave-Burgers equation with conformable derivative (cd). Some white noise functional solutions for this equation are obtained by using white noise analysis, Hermite transforms, and the modified sub-equation method. These solutions include exact stochastic trigonometric functions, hyperbolic functions solutions and wave solutions.This study emphasizes that the modified fractional sub-equation method is sufficient to solve the stochastic nonlinear equations in mathematical physics.

Invariant Solutions and Conservation Laws of the Variable-Coefficient Heisenberg Ferromagnetic Spin Chain Equation

The variable-coefficient Heisenberg ferromagnetic spin chain (vcHFSC) equation is investigated using the Lie group method. The infinitesimal generators and Lie point symmetries are reported. Four types of similarity reductions are acquired by virtue of the optimal system of one-dimensional subalgebras. Several invariant solutions are derived, including trigonometric and hyperbolic function solutions. Furthermore, conservation laws for the vcHFSC equation are obtained with the help of Lagrangian and nonlinear self-adjointness.

An useful procedure for finding exact solutions to new form of (2 + 1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation

Existence of trigonometric and hyperbolic function solutions together yields complexiton solutions. This condition is the nature of complexitons, makes the process harder and provides them a different type of wave speed. Modified double sub-equation method is a useful and practical tool to obtain complexiton solutions of nonlinear partial differential equations. Employment of two wave transformations makes modified double sub-equation method more general than methods that use one wave transformation. These two wave transformations also provide trigonometric function solutions or hyperbolic function solutions. In this study, application of modified double sub-equation method for new form of (2 + 1)-dimensional BKP equation is given. Wave propagations of some resulting solutions are illustrated.

Optical soliton solutions of the (1+1)-dimensional space-time fractional single and coupled nonlinear Schrödinger equations

The fractional double function method has been proposed to derive fractional soliton solutions and other fractional solutions of fractional nonlinear models. We find a unified solution constructed by the Mittag-Leffler function in this method, and this unified solution covers the generalized trigonometric function and hyperbolic function solutions in previous literatures. Based on this unified solution with the Mittag-Leffler function, we construct abundant solutions of the (1+1)-dimensional space-time fractional single and coupled nonlinear Schrödinger equations. From this unified solution, we find that the fractional soliton is similar to the few-cycle pulse shape and combined bright and dark soliton, and discuss their properties with the changing values of fractional order.

Complex waves and their collisions of the breaking soliton model describing hydrodynamics

As one of the important physical models, the variable-coefficient breaking soliton model describes the dynamics of solitary waves and Riemann waves in hydrodynamics. Applying the two function approach into the variable-coefficient breaking soliton model, we find an exponential-form variable separation solution with two arbitrary functions, which covers many hyperbolic and trigonometric function solutions. Based on this solution and choosing suitably two arbitrary functions, complex waves including dromion pair and single dromion superposed on a line soliton background and their collisions between complex waves and between dromion and periodic wave are discussed. For the complex waves such as the dromion pair and single dromion superposed on a line soliton background and their collisions between complex waves, two components of this model possess both striking physical meanings. However, for the collision between dromion and periodic wave, one component of this model has unique physical meaning, yet the other component of the same model appears the singularity structure. Therefore, when one discusses complex waves and their collisions for one component of the model, one must take care of non-physical structures for another component of the same model lest one arbitrarily claims that he finds the so-called novel localized structures, which are actually false non-physical structures.

Abundant solitary wave solutions to an extended nonlinear Schr\xf6dinger\u2019s equation with conformable derivative using an efficient integration method

The prevalence of the use of mathematical software has dramatically influenced the evolution of differential equations. The use of these useful tools leads to faster advances in the presentation of numerical and analytical methods. This paper retrieves several soliton solutions to the fractional perturbed Schrödinger’s equation with Kerr and parabolic law nonlinearity, and local conformable derivative. The method used in this article, called the generalized exponential rational function method, also relies heavily on the use of symbolic software such as Maple. The considered model has prominent applications in water optical metamaterials. The method retrieves several exponential, hyperbolic, and trigonometric function solutions to the model. The numerical evolution of the obtained solutions is also exhibited. The resulted wide range of solutions derived from the method proves its effectiveness in solving the model under investigation. It is also recommended to use the technique used in this article to solve similar problems.

A Variety of Novel Exact Solutions for Different Models With the Conformable Derivative in Shallow Water

For different nonlinear time conformable derivative models, a versatile built-in gadget namely the generalized exp⁡(-φ(ξ))-expansion (GEE) method is devoted to retrieving different categories of new explicit solutions. These models include the time fractional approximate long wave, the time fractional variant-Boussinesq equations and the time fractional Wu-Zhang system of equations. The GEE technique is investigated with the help of fractional complex transform and conformable derivative. As a result, we found four types of exact solutions involving hyperbolic function, periodic function, rational functional, and exponential function solutions. The physical significance of the explored solutions depends on the choice of arbitrary parameter values. Finally, we conclude that the GEE method is effective to establish the explicit new exact solutions rather than exp⁡(-φ(ξ))-expansion method.

Exploration on traveling wave solutions to the 3rd-order klein–fock-gordon equation (KFGE) in mathematical physics

In this paper, the -expansion method has been applied to find the new exact traveling wave solutions of the nonlinear evaluation equations (NLEEs) by utilizing 3rd-order Klein–Gordon Equation (KFGE). With the collaboration of symbolic commercial software maple, the competence of this method for inventing these exact solutions has been more exhibited. As an upshot, some new exact solutions are obtained and signified by hyperbolic function solutions, different combinations of trigonometric function solutions, and exponential function solutions. Moreover, the -expansion method is a more efficient method for exploring essential nonlinear waves that enrich a variety of dynamic models that arises in nonlinear fields. All sketching is given out to show the properties of the innovative explicit analytic solutions. Our proposed method is directed, succinct, and reasonably good for the various nonlinear evaluation equations (NLEEs) related treatment and mathematical physics also.Â

Analytical Solutions to the Caudrey-Dodd-Gibbon-Sawada-Kotera Equation via Symbol Calculation Approach

In this paper, we derive analytical solutions of the Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGSK) equation via symbol calculation approach. Applying the exp−φz-expansion method, we achieve the trigonometric, exponential, hyperbolic, and rational function solutions for the CDGSK equation. By choosing the appropriate parameters, we give some computer simulation to the analytical solutions of the CDGSK equation.

New generalized ϕ6-model expansion method and its applications to the (3+1) dimensional resonant nonlinear Schrödinger equation with parabolic law nonlinearity

In this paper, a new generalized ϕ6-model expansion method is proposed for the first time to construct optical solitons and other solutions of the (3+1) dimensional resonant nonlinear Schrödinger equation with parabolic law nonlinearity, which describes the propagation of optical pulse in nonlinear optical fibers. The generalized Jacobi elliptic function solutions and Weierstrass elliptic function solutions have been found. In particular, hyperbolic function solutions and periodic solutions are also obtained.

New Exact Solutions of the Conformable Space-Time Sharma–Tasso–Olver Equation Using Two Reliable Methods

The major purpose of this article is to seek for exact traveling wave solutions of the nonlinear space-time Sharma–Tasso–Olver equation in the sense of conformable derivatives. The novel ( G ′ G ) -expansion method and the generalized Kudryashov method, which are analytical, powerful, and reliable methods, are used to solve the equation via a fractional complex transformation. The exact solutions of the equation, obtained using the novel ( G ′ G ) -expansion method, can be classified in terms of hyperbolic, trigonometric, and rational function solutions. Applying the generalized Kudryashov method to the equation, we obtain explicit exact solutions expressed as fractional solutions of the exponential functions. The exact solutions obtained using the two methods represent some physical behaviors such as a singularly periodic traveling wave solution and a singular multiple-soliton solution. Some selected solutions of the equation are graphically portrayed including 3-D, 2-D, and contour plots. As a result, some innovative exact solutions of the equation are produced via the methods, and they are not the same as the ones obtained using other techniques utilized previously.

Exact traveling wave solutions to the higher-order nonlinear Schrödinger equation having Kerr nonlinearity form using two strategic integrations.

In this paper, exact traveling wave solutions to the higher-order nonlinear Schrodinger equation having Kerr nonlinearity form are derived, by adopting two relevant architecture of integration methods namely the extended sinh-Gordon equation expansion method and extended Jacobi elliptic function methods (EJEF method). The studied model, describes the propagation of optical solitons in nonlinear optical fibers. As a results, various types of traveling wave solutions are obtained including Jacobi elliptic function solutions. For some special cases, when the modulus of m approach 0 or 1 it is gained respectively periodic wave solutions and hyperbolic function solutions. Comparing our results with the well-known results in literature are also reported. Lastly the 3D profile to some of the obtained solutions are also plotted.

Computational simulations of the couple Boiti–Leon–Pempinelli (BLP) system and the (3+1)-dimensional Kadomtsev–Petviashvili (KP) equation

This research paper employs two different computational schemes to the couple Boiti–Leon–Pempinelli system and the (3+1)-dimensional Kadomtsev–Petviashvili equation to find novel explicit wave solutions for these models. Both models depict a generalized form of the dispersive long wave equation. The complex, exponential, hyperbolic, and trigonometric function solutions are some of the obtained solutions by using the modified Khater method and the Jacobi elliptical function method. Moreover, their stability properties are also analyzed, and for more interpretation of the physical features of the obtained solutions, some sketches are plotted. Additionally, the novelty of our paper is explained by displaying the similarity and difference between the obtained solutions and those obtained in a different research paper. The performance of both methods is tested to show their ability to be applied to several nonlinear evolution equations.

On the travelling wave solutions of Ostrovsky equation

In this paper, extended trial equation method (ETEM) is applied to find exact solutions of (1+1) dimensional nonlinear Ostrovsky equation. We constitute some exact solutions such as soliton solutions, rational, Jacobi elliptic and hyperbolic function solutions of this equation via ETEM. Then, we submit the results obtained by using this method.

Soliton solutions of higher-order nonlinear schrödinger equation (NLSE) and nonlinear kudryashov's equation

This article secures new soliton solutions of the higher-order nonlinear Schrödinger equation (NLSE) and the nonlinear Kudryashov's equation by means of two analytical techniques, namely the extended (G′G2)-expansion method and the first integral method. Many exact traveling wave solutions such as hyperbolic function solutions, trigonometric function solutions and rational function solutions with free parameters are characterized. Periodic solitons, dark solitons, singular solitons, combo solitons and plane waves are obtained. The existence criteria for such solutions are also provided. Moreover, the modulation instability analysis is used to examine the stabilities of both equations, by which the modulation instability gain spectrums of both equations are obtained. The obtained solutions are also presented graphically.

Singular and bright-singular combo optical solitons in birefringent fibers to the Biswas-Arshed equation

In this paper, the dynamics of optical solitons in birefringent optical fibers without four-wave mixing is addressed with the advocacy of Biswas-Arshed equation in media with Kerr type nonlinearity. Singular solitons and bright-singular combo optical solitons are obtained by using exp-function method. Variety of other traveling wave solutions are also formulated with this integration scheme. These solutions include rational function solutions, mixed hyperbolic function solutions and combination of trigonometric function solutions. Some restraint conditions pop out naturally from the mathematical analysis that must be gratified to guarantee the existence of solitons. Moreover, for the physical illustration of the obtained solutions, 3D and 2D graphs are presented.