Trial Equation Method Research Articles

Optical envelope patterns in quadratic–cubic nonlinear medium by trial equation method

In nonlinear optics, different forms of soliton propagation can be described by nonlinear Schrödinger equation. The purpose of this paper is to find and classify all exact solutions of a class of Schrödinger equation in quadratic-cubic nonlinear medium according to the quartic polynomial discriminant system and trial equation method. These solutions contain the partial solutions obtained by predecessors with different algorithms. The complete discriminant system of quartic polynomials is modified, and the eight families of exact solutions show various properties of patterns of amplitudes including singularity, periodicity and double periodicity.

Optical envelope patterns perturbation with full nonlinearity for Gerdjikov–Ivanov equation by trial equation method

In this paper, the classification of all envelope traveling wave solutions of Gerdjikov–Ivanov model with full nonlinearity is obtained by using complete discriminant system for polynomial method and trial equation method. All the results show that the Gerdjikov–Ivanov model has abundant traveling wave solutions, including isolated solutions, periodic solutions, rational singular solutions, double periodic continuous solutions and discontinuous solutions. Among them, some new solutions are obtained.

Dynamics of combined soliton solutions of unstable nonlinear fractional-order Schrodinger equation by beta-fractional derivative

In this article, a new version of the trial equation method is suggested. This method allows new exact solutions of the nonlinear partial differential equations. The developed method is applied to unstable nonlinear fractional-order Schr¨odinger equation in fractional time derivative form of order. Some exact solutions of the fractional-order fractional PDE are attained by employing the new powerful expansion approach using by beta-fractional derivatives which are used to get many solitary wave solutions by changing various parameters. New exact solutions are expressed with rational hyperbolic function solutions, rational trigonometric function solutions, 1-soliton solutions, dark soliton solitons, and rational function solutions. We can say that the unstable nonlinear Schr¨odinger equation exists I different dynamical behaviors. In addition, the physical behaviors of these new exact solution are given with two and three dimensional graphs.

The Modified Trial Equation Method to the Van der Waals Model

In this research, the modified trial equation method (MTEM) is considered in order to find some exact solutions of the van der Waals model. In addition, to finding the solution of the van der Waals model this method can be used in the solution of nonlinear problems. Thus, some wave solutions for various situations are obtained. Also, three and two dimensional graphs were found with the help of Mathematica9 to analyze the physical behavior of the obtained solutions.

Two model equations with a second degree logarithmic nonlinearity and their Gaussian solutions

In the paper, we try to study the mechanism of the existence of Gaussian waves in high degree logarithmic nonlinear wave motions. We first construct two model equations which include the high order dispersion and a second degree logarithmic nonlinearity. And then we prove that the Gaussian waves can exist for high degree logarithmic nonlinear wave equations if the balance between the dispersion and logarithmic nonlinearity is kept. Our mathematical tool is the logarithmic trial equation method.

Exact solutions of non-linear Klein–Gordon equation with non-constant coefficients through the trial equation method

In this note, we use an extension of the trial equation method (also called the direct integral method) for partial differential equations with non-constant coefficients to derive exact solutions in the form of nonlinear waves. The model considered generalizes other classical models from physics like the Klein–Gordon equation, the $$(1 + 1)$$ -dimensional $$\phi ^4$$ -theory, the Fisher–Kolmogorov equation from population dynamics, and the Hodgkin–Huxley model which describes the propagation of electrical signals through the nervous system. As a particular example, the cylindrically symmetric cubic nonlinear Klein–Gordon equation is considered herein.

Stationary envelope solutions of a nonlinear Schrödinger-type equation

By using complete discrimination system for polynomial method and trial equation method, a series of stationary envelope patterns for a nonlinear schrödinger-type equation are given, which conclude solitary waves pattern, double periodic patterns, soliton pattern and so on. And we draw the figures which help us to understand the dynamics behaviors of the nonlinear schrödinger-type equation intuitionally.

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 optical solitons of conformable resonant nonlinear Schrödinger’s equation

AbstractSardar subequation approach, which is one of the strong methods for solving nonlinear evolution equations, is applied to conformable resonant Schrödinger’s equation. In this technique, if we choose the special values of parameters, then we can acquire the travelling wave solutions. We conclude that these solutions are the solutions obtained by the first integral method, the trial equation method, and the functional variable method. Several new traveling wave solutions are obtained including generalized hyperbolic and trigonometric functions. The new derivation is of conformable derivation introduced by Atangana recently. Solutions are illustrated with some figures.

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.

Dynamics of combined soliton solutions of unstable nonlinear Schrodinger equation with new version of the trial equation method

In this article, a new version of the trial equation method is suggested. With this method, it is possible to find the new exact solutions of the nonlinear partial differential equations. The developed method is applied to unstable nonlinear Schrödinger equation. New exact solutions are expressed with Jacobi elliptic function solutions, 1-soliton solutions and rational function solutions. When the obtained results are examined, we can say the unstable nonlinear Schrödinger equation shows different dynamic behaviors. In addition, the physical behaviors of these new exact solution are given with two and three dimensional graphs.

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.

Optical propagation patterns in nonlocal parabolic law medium

The model of optical propagation in nonlocal parabolic law medium is described by nonlinear Schrödinger equation with high-order nonlinear terms. Exact optical propagation patterns are constructed by proposing a general complex trial equation method. These results show rich optical propagation patterns of the model.

Optical solitons and envelop propagation patterns in birefringent polarization-preserving fiber

The optical propagation in the birefringence polarization-preserving fiber is modeled by the nonlinear Schrödinger equations system including the effect of four-wave mixing. A general trial equation method for complex coupled system is introduced to give the optical solitons and envelope solutions. These rich results show a varied of optical propagation patterns of the model. Under the concrete parameters, we give representations of these optical propagation patterns.

Soliton solutions of coupled higgs field equation via the trial equation method

Since a few recent decades, investigation of nonlinear evolution equations (NLEEs) is becoming an important area of research as they have a variety of applications in various branches of social and scientific disciplines like Ecology, Social Dynamics, Financial Mathematics, Engineering and many branches of Physics such as Biophysics, Chemical Physics, Fibre Optics, Fluid Mechanics, Neuro-physics, Particle Physics, Solid State Physics and many more. Many powerful and efficient methods of finding exact solutions of NLEEs have been proposed so far and the Trial Equation Method [ 1 - 5] is one of them. Many authors have successfully used the method in finding exact solutions of a number of NLEEs. In the present paper, soliton solutions of the Coupled Higgs Field Equation [ 6 - 10 ] are being obtained using the Trial Equation Method. The Coupled Higgs Field Equation describes system of conserved scalar nucleons interacting with neutral scalar mesons in particle physics. This coupled equation has applications in the studies of Field Theory and Electromagnetic waves as well. This coupled equation introduces the Higgs field to illustrate the mechanism of generation of mass for Gauge Bosons. The Coupled Higgs Field Equation is generally expressed as the following pair of NLEEs                                                                                                                                                          (3) and                                                                                                                                                                          (2) Here, x and t are spatial and temporal variables respectively, the function  is a complex scalar nucleon field, the function  is a real scalar meson field,  are arbitrary real constants and the subscripts denote partial differentiations with respect to them.Using the Trial Equation Method, the above coupled NLEE is to be solved to obtain some soliton solutions.

Exact dynamical behavior for a dual Kaup\u2013Boussinesq system by symmetry reduction and coupled trial equations method

We propose a coupled trial equation method for a coupled differential equations system. Furthermore, according to the invariant property under the translation, we give the symmetry reduction of a dual Kaup–Boussinesq system, and then we use the proposed trial equation method to construct its exact solutions which describe its dynamical behavior. In particular, we get a cosine function solution with a constant propagation velocity, which shows an important periodic behavior of the system.

Conservation laws, analytical solutions and stability analysis for the time-fractional Schamel\u2013Zakharov\u2013Kuznetsov\u2013Burgers equation

In this paper, we consider the (3+1)-dimensional time-fractional Schamel–Zakharov–Kuznetsov–Burgers (SZKB) equation. With the help of the Riemann–Liouville derivatives, the Lie point symmetries of the (3+1)-dimensional time-fractional SZKB equation are derived. By applying the Lie point symmetry method as well as Erdélyi–Kober fractional operator, we get the similarity reductions of the time-fractional SZKB equation. Conservation laws of the time-fractional SZKB are constructed. Moreover, we obtain its power series solutions with the convergence analysis. In addition, the analytical solution is obtained by modified trial equation method. Finally, stability is analyzed graphically in different planes.

Optical envelop patterns in quadratic-cubic nonlinear medium by complex trial equation method

Exact envelop patterns of nonlinear Schrödinger equation (NLSE) with perturbation terms are constructed to describe optical solitons in quadratic-cubic nonlinear medium by proposing a complex version of Liu’s trial equation method. These four families of envelop solutions show various properties of patterns of amplitudes including singularity, periodicity and double periodicity.

Exact single traveling wave solutions to generalized (2+1)-dimensional Gardner equation with variable coefficients

We firstly apply the trial equation method to generalized (2+1)-dimensional Gardner equation to reduce the nonlinear partial differential equation into ordinary equation. Then by the complete discrimination system for polynomial method, the classification of the exact single traveling wave solutions is presented. This is the first time that the two methods are applied to integral-differential equation. In the classification, we can see that the equation has rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions solutions with double periodic, which are impossible to be obtained by other methods. Moreover, to ensure the existences of these solutions, concrete examples are also constructed with specific parameters. To the best of our knowledge, the results presented in the paper are brand new and can not be found in any other papers.