Trigonometric Function Solutions Research Articles

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.

Complex traveling-wave and solitons solutions to the Klein-Gordon-Zakharov equations

This paper studies complex solutions and solitons solutions to the Klein-Gordon-Zakharov equations (KGZEs). Solitons solutions including bright, dark, W-shape bright, breather also trigonometric function solutions and singular solutions of KGZEs are obtained by three integration algorithm. From the spatio-temporal and 3-D and 2-D contour plot, it is observed that obtained solutions move without any deformation that implies the steady state of solutions. Furthermore, these solutions will be helpful to explain the interactions in hight frequency plasma and solitary wave theory.

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.

Application of the modified exponential function method to Vakhnenko-Parkes equation

In this paper, we submit some new travelling wave solutions for the Vakhnenko--Parkes equation via the modified exponential function method. The obtained solutions include hyperbolic, exponential, trigonometric function solutions. Regarding these solutions, the 2D and 3D graphs and contour simulations are presented.

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.

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.

Optical dark and singular solitons to the Biswas-Arshed equation in birefringent fibers without four-wave mixing

This paper addresses the dynamics of optical solitons in birefringent fibers with the advocacy of Biswas-Arshed equation without four-wave mixing terms, in media with Kerr-law nonlinearity. Dark and singular optical solitons are formulated by utilizing a resourceful integration methodology, the extended (G′/G)-expansion method. Variety of other traveling wave solutions are also formulated with this integration technique. These solutions include rational function solutions, mixed hyperbolic function solutions and combination of trigonometric function solutions. Some constraint relations are also listed which emerged from the mathematical analysis and must be pleased to guarantee the existence of solitons.

Bright-Dark and Multi Solitons Solutions of (3 + 1)-Dimensional Cubic-Quintic Complex Ginzburg-Landau Dynamical Equation with Applications and Stability.

In this paper, bright-dark, multi solitons, and other solutions of a (3 + 1)-dimensional cubic-quintic complex Ginzburg–Landau (CQCGL) dynamical equation are constructed via employing three proposed mathematical techniques. The propagation of ultrashort optical solitons in optical fiber is modeled by this equation. The complex Ginzburg–Landau equation with broken phase symmetry has strict positive space–time entropy for an open set of parameter values. The exact wave results in the forms of dark-bright solitons, breather-type solitons, multi solitons interaction, kink and anti-kink waves, solitary waves, periodic and trigonometric function solutions are achieved. These exact solutions have key applications in engineering and applied physics. The wave solutions that are constructed from existing techniques and novel structures of solitons can be obtained by giving the special values to parameters involved in these methods. The stability of this model is examined by employing the modulation instability analysis which confirms that the model is stable. The movements of some results are depicted graphically, which are constructive to researchers for understanding the complex phenomena of this model.

Applications of three methods for obtaining optical soliton solutions for the Lakshmanan–Porsezian–Daniel model with Kerr law nonlinearity

This paper examines new travelling wave solutions to the Lakshmanan–Porsezian–Daniel (LPD) model with Kerr nonlinearity using Backlund transformation method based on Riccati equation, Kudryashov method and a new auxiliary ordinary differential equation (ODE). The three methods are adequately utilised, and some new rational-type hyperbolic and trigonometric function solutions are derived in different shapes for the aforementioned model. We confirm that our methods are more efficient than the other methods and it might be used in many other such types of nonlinear equations arising in the basic fabric of communications network technology and nonlinear optics.

On new computational and numerical solutions of the modified Zakharov–Kuznetsov equation arising in electrical engineering

In this research, analytical and numerical solutions are studied of a two–dimensional discrete electrical lattice, which is mathematically represented by the modified Zakharov–Kuznetsov equation. Moreover, the stability property of the obtained analytical solutions is investigated based on the Hamiltonian system, and then it is used to evaluate the initial and boundary conditions that are used in the numerical investigation. Many kinds of analytical solutions are obtained, such as complex, exponential, hyperbolic, and trigonometric function solutions. The functioning of both schemes of both techniques is tested and investigated.

Two different systematic methods for constructing meromorphic exact solutions to the KdV-Sawada-Kotera equation

In this paper, we obtain meromorphic exact solutions of the KdV-Sawada-Kotera equation via two different systematic methods. Applying the exp(-ψ(z))-expansion method, we achieve the trigonometric, exponential, hyperbolic and rational function solutions for the mentioned equation. It is more interesting that we firstly proposed the extended complex method based on the previous work of Yuan et al., and as an example we use it to search exact solutions to the KdV-Sawada-Kotera equation. Dynamic behaviors of solutions obtained by these two different systematic techniques are also shown by some graphs. The results show that these two methods are direct and efficient methods to deal with various differential equations in the applied sciences.

Constructing new wave solutions to the $$(2 + 1)$$-dimensional Davey–Stewartson equation (DSE) which arises in fluid dynamics

In this paper, we utilize the uniform algebraic method and construct some new wave solutions to the $$(2 + 1)$$-dimensional Davey–Stewartson equation which arises in fluid dynamics. We successfully obtain some hyperbolic and trigonometric function solutions to this equation. Subsequently, We have also built the Lagrangian and the Hamiltonian for the second-order nonlinear ODE corresponding to the traveling wave reduction of the (DSE)-equation. Moreover, we plot 2D and 3D surfaces representing the obtained solutions by considering some suitable values of the parameters. The unique (2 + 1)-dimensional dynamics of the obtained exact solutions of the (DSE) equation may help to understand the formation and key properties of traveling waves in many other physical settings. Before the end of this paper, we compare our results with the existing results in the literature by presenting comprehensive conclusions.

Analytical wave solutions of the (2 + 1)-dimensional first integro-differential Kadomtsev-Petviashivili hierarchy equation by using modified mathematical methods

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.

Bifurcations and solutions for the generalized nonlinear Schrödinger equation

Abstract The theory of bifurcations for dynamical system is employed to construct new exact solutions of the generalized nonlinear Schrodinger equation. Firstly, the generalized nonlinear Schrodinger equation was converted into ordinary differential equation system by using traveling wave transform. Then, the system's Hamiltonian, orbits phases diagrams are found. Finally, six families of solutions are constructed by integrating along difference orbits, which consist of Jacobi elliptic function solutions, hyperbolic function solutions, trigonometric function solutions, solitary wave solutions, breaking wave solutions, and kink wave solutions.

Symmetry reduction, exact solutions and conservation laws of the Bogoyavlenskii equation

In this paper, by applying the direct symmetry method, we obtain the symmetry reductions, group invariant solutions and some new exact solutions of the Bogoyavlenskii equation, which include hyperbolic function solutions, trigonometric function solutions and power series solutions. We also give the conservation laws of the Bogoyavlenskii equation.

Dispersive solitary wave and soliton solutions of the gernalized third order nonlinear Schrödinger dynamical equation by modified analytical method

In this article, Generalized Riccati mapping method is analysed to drive the several sorts of solitons such as dark and bright solitons, periodic solitons, trigonometric function and solitary waves solutions of generalized non-linear Schrödinger equation of third order. This dynamical model has significance application in various fields such as it is utilized in optical fibers to model pulses of ultra-short. We also present graphical presentation of various types of solution in 3 dimensional, 2 dimensional and counter plot to make understand the physical phenomena’s. The computing exertion and attained results confirm the influence and effectiveness of current methods.

Application of the extended exp(-φ(ξ))-expansion method to the nonlinear conformable time-fractional partial differential equations

This paper investigates the new exact solutions of the three nonlinear time fractional partial differential equations namely the nonlinear time fractional Clannish Random Walker’s Parabolic (CRWP) equation, the nonlinear time fractional modified Kawahara equation, and the nonlinear time fractional BBM-Burger equation by utilizing an extended form of exp(-φ(ξ))-expansion method in the sense of conformable fractional derivative. As outcomes, some new exact solutions are obtained and signified by hyperbolic function solutions, trigonometric function solutions, and rational function solutions. Some solutions have been plotted by MATLAB software to show the physical significance of our studied equations. In the point of view of our executed method and generated results, we may conclude that extended exp (-φ(ξ))-expansion method is more efficient than exp(-φ(ξ))-expansion method to extract the new exact solutions for solving any types of integer and fractional differential equations arising in mathematical physics.  Â

Searching for Analytical Solutions of the (2+1)-Dimensional KP Equation by Two Different Systematic Methods

In this paper, we derive analytical solutions of the (2+1)-dimensional Kadomtsev-Petviashvili (KP) equation by two different systematic methods. Using the exp⁡(-ψ(z))-expansion method, exact solutions of the mentioned equation including hyperbolic, exponential, trigonometric, and rational function solutions have been obtained. Based on the work of Yuan et al., we proposed the extended complex method to seek exact solutions of the (2+1)-dimensional KP equation. The results demonstrate that the applied methods are efficient and direct methods to solve the complex nonlinear systems.

New exact solutions for Chan-Hilliard equation by new sub-equation method

In this present study, new sub-equation method is used to construct exact travelling wave solutions of the Cahn-Hilliard equation (CH). As a consequence, many new general exact travelling wave solutions are obtained including generalized hyperbolic (GH) function, generalized trigonometric (GT) function, exponential function and rational function solutions. Compared with the most existing similar new sub-equation method, the proposed method gives new and more general exact travelling wave solutions. More importantly, the method with the help of symbolic computation prepares a powerful mathematical tool for solving a great many nonlinear evolution equations in mathematical physics.