Extended Trial Equation Method Research Articles

Exact Periodic Wave Solutions for the Perturbed Boussinesq Equation with Power Law Nonlinearity

In this paper, exact periodic wave solutions for the perturbed Boussinesq equation with power law nonlinearity are obtained for different nonlinear strengths n. When n=1, the periodic traveling wave solutions can be found by the definition of the Jacobian elliptic function. When n≥1, we construct a transformation to solve for the power law nonlinearity, and the periodic traveling wave solutions can be obtained by applying the extended trial equation method. In addition, we consider the limiting case where the periodicity of the periodic traveling wave solutions vanishes, and we obtain the soliton solution for n=1. Numerical simulations show the periodicity of the solution for the perturbed Boussinesq equation.

Exact traveling wave solutions of (2+1)-dimensional extended Calogero–Bogoyavlenskii–Schiff equation using extended trial equation method and modified auxiliary equation method

Exact traveling wave solutions of (2+1)-dimensional extended Calogero–Bogoyavlenskii–Schiff equation using extended trial equation method and modified auxiliary equation method

The traveling wave solutions to a variant of the Boussinesq equation

In this study, we study a variant of the Boussinesq equation called as \(B( n+1,\,1,\,n )\) equation, and construct some traveling wave solutions by using an effective approach called the extended trial equation method. Thus, the soliton solutions, rational function solutions, elliptic function solutions and Jacobi elliptic function solutions, which show the existence of various mathematical and physical structures and events in the fundamental equation considered, have been constructured. In order to make a more detailed examination of the physical behavior of these solutions, two- and three-dimensional graphs of some solution functions were drawn with the help of the Mathematica package program. In the section of Discussion, we suggest a more general version of the trial equation method for nonlinear differential equations.

Optical soliton solutions of the generalized sine-Gordon equation

In this study, the extended trial equation method based on the general form of the nonlinear elliptic ordinary differential equation isemployed to solve the nonlinear generalized sine-Gordon equations. By using this method, we achieve, unlike new types of exact wave solutions such as Elliptic-F, Elliptic-E, and Elliptic-Π functions that are known as elliptic integrals.

On Travelling Wave Solutions of Dullin-Gottwald-Holm Dynamical Equation and Strain Wave Equation

In this study, extended trial equation method (ETEM) is implemented to obtain exact solutions of the Dullin-Gottwald-Holm Dynamical equation (DGHDE) and the strain wave equation. We constitute some exact solutions such as soliton solutions, rational, Jacobi elliptic, periodic wave solutions and hyperbolic function solutions of these equations via ETEM. Then, we present results that we obtained by using this method.

Exact solutions of the combined Hirota-LPD equation with variable coefficients

In this paper, we construct exact families of traveling wave (periodic wave, singular wave, singular periodic wave, singular-solitary wave and shock wave) solutions of a well-known equation of nonlinear PDE, the variable coefficients combined HirotaLakshmanan-Porsezian-Daniel (Hirota-LPD) equation with the fourth nonlinearity, which describes an important development, and application of soliton dispersion management experiment in nonlinear optics is considered, and as an achievement, a series of exact traveling wave solutions for the aforementioned equation is formally extracted. This nonlinear equation is solved by using the extended trial equation method (ETEM) and the improved tan(ϕ/2)-expansion method (ITEM). Meanwhile, the mechanical features of some families are explained through offering the physical descriptions. Analytical treatment to find the nonautonomous rogue waves are investigated for the combined Hirota-LPD equation.

Extraction of optical solitons in birefringent fibers for Biswas-Arshed equation via extended trial equation method

Abstract This article obtains optical solitons to the Biswas-Arshed equation for birefringent fibers with higher order dispersions and in the absence of four-wave mixing terms, in a media with Kerr type nonlinearity. Optical dark, singular and bright soliton solutions are articulated by applying an imaginative integration technique, the extended trial equation scheme. Various additional traveling wave solutions are produced with this integration technique, which include rational solutions, Jacobi elliptic function solutions and periodic singular solutions. From the mathematical analysis some constraints are recognized that ensure the actuality of solitons.

New Exact Solutions of (3+1)-dimensional Kadomtsev-Petviashvili (KP) Equation

The extended trial equation method is investigated which allows us to achieve soliton solutions and Jacobi elliptic function solution of the partial differential equations. This method is implemented to the (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation and various new exact solutions have been obtained. These new obtain exact solutions are solutions that are not known in the literature. Additionally, two and three-dimensional graphics were drawn to understand the physical behaviors of the distinct obtain exact solutions.

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.

Dynamics of optical solitons incorporating Kerr dispersion and self-frequency shift

This paper deals with the dynamics of optical solitons in nonlinear Schrödinger equation (NLSE) with cubic-quintic law nonlinearity in the presence of self-frequency shift and self-steepening. This type of equation describes the ultralarge capacity transmission and traveling of laser light pulses in optical fibers. Two robust analytical approaches are employed to determine contemporary solutions. Some new explicit rational, periodic and combo periodic soliton solutions are extracted using the extended trial equation method. The Riccati–Bernoulli sub-ODE method provided us with singular and dark soliton solutions. The constraints found are necessary for the existence of solitons.

Optical solitons for paraxial wave equation in Kerr media

This paper retrieves Jacobi elliptic, periodic, bright and singular solitons for paraxial nonlinear Schrödinger equation (NLSE) in Kerr media. We use extended trial equation method to obtain these solitons solutions. For the existence of the soliton solutions, constraint conditions are also presented.

Exact Solutions for a Class of Nonlinear PDE with Variable Coefficients Using ET and ETEM

In this article, by using the modified CK direct method, we give a relationship between the generalized fifth-order KDV equations with variable coefficients and the corresponding constant coefficients ones. Then, we construct the abundant travelling solutions by the extended trial equation method (ETEM) in terms of different functions, such as the elliptic functions, rational functions, hyperbolic functions and trigonometric functions. The extended trial equation method is powerful and can be used to other partial differential equations and more research can be done by this method.

Analytical treatments of the space–time fractional coupled nonlinear Schrödinger equations

Under investigation in this work is a ( $$2+1$$ )-dimensional the space–time fractional coupled nonlinear Schrodinger equations, which describes the amplitudes of circularly-polarized waves in a nonlinear optical fiber. With the aid of conformable fractional derivative and the fractional wave transformation, we derive the analytical soliton solutions in the form of rational soliton, periodic soliton, hyperbolic soliton solutions by four integration method, namely, the extended trial equation method, the $$\exp (-\,\Omega (\eta ))$$ -expansion method and the improved $$\tan (\phi (\eta )/2)$$ -expansion method and semi-inverse variational principle method. Based on the the extended trial equation method, we derive the several types of solutions including singular, kink-singular, bright, solitary wave, compacton and elliptic function solutions. Under certain condition, the 1-soliton, bright, singular solutions are driven by semi-inverse variational principle method. Based on the analytical methods, we find that the solutions give birth to the dark solitons, the bright solitons, combine dark-singular, kink, kink-singular solutions with fractional order for nonlinear fractional partial differential equations arise in nonlinear optics.

Complementary wave solutions for the long-short wave resonance model via the extended trial equation method and the generalized Kudryashov method

Abstract In this paper the nonlinear long-short (LS) wave resonance model is analyzed through a new perspective. We obtain the classification of exact solutions by making use of the complete discrimination system for the trial equation method and through the generalized Kudryashov method. These methods do generate complementary wave solutions such as bright and dark solitons, rational functions, Jacobi elliptic functions as well as singular and periodic wave solutions. Some among them extend the already reported solutions through other techniques. For some types of solutions adequately graphical representations are displayed. The concerned methods could also be used in order to study other interesting nonlinear evolution processes in n dimensions.

Abundant travelling wave solutions for nonlinear Kawahara partial differential equation using extended trial equation method

ABSTRACTIn this paper, we use the extended trial equation method (ETEM) to construct the exact solutions in many different functions such as the trigonometric function, elliptic integral function, rational function, hyperbolic function and Jacobi elliptic functions for nonlinear evolution equation in mathematical physics via the fifth-order modified nonlinear Kawahara equation. In this method, the balance number is not constant as we have shown in other methods, but it is changed by changing the trial equation derivative definition. This method is powerful, reliable, effective and simple for solving more complicated nonlinear partial differential equations in mathematical physics.

Solitons in optical metamaterials with anti-cubic law of nonlinearity by ETEM and IGEM

BackgroundThis paper studies soliton perturbation in optical metamaterials, with anticubic nonlinearity.MethodsTwo integration approaches, namely, the extended trial equation method (ETEM) and the improved G’/Gexpansion method (IGEM) are presented.ResultsBright, dark and singular soliton solutions are retrieved. The existence criteria of these solitons in metamaterials are also demonstrated. All solutions have been verified back into its corresponding equation with the aid of maple package program.ConclusionsFinally, we believe that the executed method is robust and efficient than other methods and the obtained solutions in this paper can help us to understand the variation of solitary waves in optical metamaterials.

Optical solitons in birefringent fibers with weak non-local nonlinearity and four-wave mixing by extended trial equation method

This paper studies optical solitons in birefringent fibers with parabolic law nonlinearity coupled with waekly non-local nonlinear medium. The four-wave mixing effect is taken into consideration and thus retrieval of soliton solutions would require phase-matching condition.

Optical dark and dark-singular soliton solutions of (1+2)-dimensional chiral nonlinear Schrodinger’s equation

ABSTRACTIn this study, the dynamical analysis of optical dark and singular solitons is carried out for chiral (1+2)-dimensional nonlinear Schrödinger’s equation with the implementation of extended direct algebraic and extended trial equation method independently. The constraint conditions guarantee the perseverance of these soliton solutions. Along with optical dark and singular solitons, these integration techniques yield other wave solutions such as Jacobi elliptic function, rational function, and hyperbolic function solutions as outgrowth.

New optical solitons of Tzitzeíca type evolution equations using extended trial approach

The properties of Tzitzeica equations in nonlinear optics have been the discussion of many recent studies. In this article, a new and productive implementation of trial equation method is chosen to handle this class of nonlinear evolution equations. As a result, the new and more exact periodic, singular, hyperbolic and rational solutions of Tzitzeica, Dodd–Bullough–Mikhailov and Tzitzeica–Dodd–Bullough equations are formally obtained. The extended trial equation method along with the symbolic computation gives a versatile technique to deal such kinds of nonlinear evolution equations in nonlinear optics.

Exact optical solitons in (n + 1)-dimensions with anti-cubic nonlinearity

The paper studies the propagation of optical solitons in [Formula: see text]-dimensions under anti-cubic law of nonlinearity. The bright, dark and singular optical solitons are extracted using the extended trial equation method. The constraint conditions, for the existence of these solitons, are also listed. Additionally, a couple of other solutions known as singular periodic and Jacobi elliptic solutions, fall out as a by-product of this scheme. The obtained results are new and reported first time in [Formula: see text]-dimensions with anti-cubic law of nonlinearity.