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Trial equation method and its applications to nonlinear evolution equations
Abstract
A new method, that is, trial equation method, was given to obtain the exact trav eling wave solutions for nonlinear evolution equations. As an example, a class o f fifth-order nonlinear evolution equations was discussed. Its exact traveling w ave solutions, which included rational form solutions, solitary wave solutions, triangle function periodic solutions, polynomial type Jacobian elliptic function periodic solutions and fractional type Jacobian elliptic function periodic solu tions, were given.
... The intermediary functions that made these soliton solutions retrieval possible are Jacobi's elliptic functions, with the limiting value of the modulus of ellipticity approaching unity. Also, we disregard additional solutions, including singular periodic solutions and plane waves, that arise from alternate signs of the discriminant [22][23][24][25][26], as they are irrelevant to the context of optoelectronics. The results and their derivations are exhibited in the rest of the paper after a quick introduction to the model. ...
... The complete discriminant approach was to the rescue. We ignore the emergence of singular periodic solutions and plane waves due to alterations in the discriminant's signs [22][23][24][25][26], as they hold no significance in optoelectronics. Thus, from the optics perspective, a complete spectrum of optical solitons has been recovered using the complete discriminant approach and is being reported in this paper. ...
... Later, the model will be numerically addressed using the Laplace-Adomian decomposition approach and/or variational iteration method. The results of such research activities will be sequentially disseminated after they are all connected with the pre-existing ones [22][23][24][25][26]. ...
... By virtue of the usability of the third-order polynomial complete discriminant system [68][69][70], one considers the decomposition of 3D-HSII system (7) in the suceeding four cases, see [71]. In addition, we present the outcomes in the Families of solutions [71]. ...
... By virtue of the usability of the third-order polynomial complete discriminant system [68][69][70], one considers the decomposition of 3D-HSII system (7) in the suceeding four cases, see [71]. In addition, we present the outcomes in the Families of solutions [71]. In the first place, we consider the situation when = 0 and D 1 < 0: Case 1. ...
Models reveal the dynamic character of wave motion, delineated in shallow waters alongside fluid dynamics; an example is the Hirota–Satsuma–Ito model equation. Therefore, this paper showcases the detailed analytical investigations of a (3+1)-dimensional Hirota–Satsuma–Ito-like system. In order to reduce the equation to a nonlinear ordinary differential system of equations, a traveling plane wave transformation is engaged. Thereafter, the direct integration technique is adopted to solve the model, thus culminating in obtaining Jacobi elliptic integral function solutions. Moreover, to attain more various solitonic solutions of diverse structures, a standard approach called the polynomial complete discriminant system and elementary integral technique is engaged. This provides exact traveling wave solutions of diverse known functions in the form of periodic, trigonometric, dark, mixed bright, and topological kink, as well as singular soliton solutions. These are found to appear in the form of Jacobi elliptic, trigonometric, as well as hyperbolic functions. Furthermore, some of these solutions are further examined by investigating their wave nature via numerical simulations.
... In this paper, we aim to expand the categories of solutions to provide broader theoretical support. This paper chiefly applies the complete discrimination system for polynomial method (CDSPM) [37,38] and the trial equation method [39] by Liu to gain complete classification of all single traveling wave solutions and provide the topological stability analysis of them. Besides, based on the results of Kai and Hu [40][41][42], we conduct a qualitative analysis of the equation, which expounds that periodic solutions and soliton solutions are subsistent. ...
... Now, we establish the trial equation [39] ( ...
The cubic–quartic perturbed Biswas–Milovic equation, which contains Kudryashov's nonlinear form and two generalized nonlocal laws, has been explored qualitatively and quantitatively, as demonstrated in the present work. The research methods used include the complete discrimination system for polynomial method and the trial equation method. The results show that the Hamiltonian has the conservation property, and the global phase diagrams obtained via the bifurcation method reveal the existence of periodic and soliton solutions. Furthermore, we fully classify all the single traveling wave solutions to substantiate our findings, covering singular solutions, solitons, and Jacobian elliptic function solutions. We analyze their topological stabilities and present two‐dimensional graphs of solutions. We also delve deeper into the dynamic system by incorporating the perturbation item to explore the chaotic phenomena associated with the equation. These outcomes are valuable for studying the propagation of high‐order dispersive optical solitons and have potential applications in optimizing optical communication systems to improve efficiency.
... The current paper is a study of this model in the presence of perturbation terms. The integration methodology is the complete discriminant classification approach [16][17][18][19][20][21][22]. The governing model is first transformed into an ordinary differential equation (ODE), which is subsequently integrated based on the structural classification of the corresponding discriminant. ...
... Next, we give the discriminant system [16][17][18][19][20][21][22]: ...
... However, these approaches have limitations, and the single traveling wave solutions that they produced are incomplete. In this paper, we apply the trial equation method [34] and the complete discrimination system for polynomial method [35,36] by Liu to solve the Biswas-Milovic equation with nonlinear perturbation terms. We investigate the dynamic behaviors of the equation, carry out qualitative analysis and quantitative, and verify the existence of solitons and periodic solutions. ...
... Take the trial equation [34] as ...
This article mainly explores the Biswas–Milovic equation containing Kudryashov's refractive index law and nonlinear perturbation terms. We take the complete discrimination system for polynomial method and the trial equation method to implement qualitative and quantitative studies for the equation. We verify that the Hamiltonian of the dynamic system is conserved and apply the complete discrimination system of polynomial method to give the associated global phase diagrams, which are qualitatively analyzed to illustrate the presence of periodic solutions and solitons. In addition, we gain a classification of all single traveling wave solutions, provide optical wave patterns with specific parameters and validate the above findings.
... Shallow water or long water wave equation is a special kind of equation describing wave propagation with the wavelength much longer than its depth. Due to the wide applications in many fields of science and engineering such as harbor designing and electrical circuits [1,2], this kind of equations has attracted a lot attention, and many famous methods have been proposed to obtain its exact solutions, for example, CDSPM [3][4][5]], trial equation method [6][7][8], and so on. Among all kinds of exact solutions, solitary wave solution plays a vital role, because it is related to many phenomena arisen in physics [9]. ...
... Equation (29) could be regarded as a special kind of trial equation. Different from the traditional trial equation [6][7][8] setting ...
A generalization of the regularized long-wave equation is considered, and the existences of smooth soliton, peakon, and periodic solutions are established via the complete discrimination system for polynomial method and the bifurcation method. Concrete examples of these solutions are constructed to verify our conclusions directly. In particular, we construct a special kind of smooth soliton solution, namely a Gaussian soliton solution, and give two sufficient conditions for the existence of such a solution by the extended trial equation method. To the best of our knowledge, this is the first time that a Gaussian soliton solution has been constructed for an equation with no logarithmic nonlinearity.
... In the literature, there are various methods for obtaning the numerical or analytical solutions of such equations. Some of these methods in the literatüre are, the generalized Bernoulli sub-equation function method [1][2], the trial equation method [3][4][5][6][7], the modified, extended tanh-function method [8][9], the first integral method [10], generalized tanh function method [11], the modified exponential function method [12][13][14][15][16] and many more methods. ...
... If UV transform is applied in order to make integral operations with ease in equation (12), If 1 M is chosen so as to provide the equality in equation (14), 3 N is obtained. In this case, the necessary derivative terms in equation (5) and the nonlinear ordinary differential equation are obtained as follows: Figure 2. Three-dimensional, contour and density plots of solution (19) for the values 3 ...
Bu çalışmada, (3 + 1) boyutlu potansiyel Yu-Toda-Sasa-Fukuyama (YTSF) denkleminin hareketli dalga çözümleri, modifiye üstel fonksiyon yöntemi (MEFM) kullanılarak elde edilmiştir. Bulunan çözüm fonksiyonları incelendiğinde trigonometrik, hiperbolik ve rasyonel fonksiyonların olduğu görülmektedir. Elde edilen çözüm fonksiyonları, (3 + 1) boyutlu potansiyel Yu-Toda-Sasa-Fukuyama (YTSF) denklemini sağlayan Wolfram Mathematica yazılımı ile kontrol edildi. Uygun parametreler belirlenerek çözüm fonksiyonunun iki ve üç boyutlu ve kontur grafikleri bulundu.
... Thus, we use the trial equation method to find its integrable factor equation from which we get its Gaussian solution. The trial equation method is a useful mathematical tool for solving some higher-order nonlinear differential equations [27,[31][32][33][34][35][36]. ...
... We must prove that there exist concrete physical parameters so that we can realize the concrete Gaussian solitary waves. Indeed, from the expressions of parameters in (21), and denoting q = 2|1−α| l 2 30 mω 2 , we have from (30) and (31) ...
We consider a Fermi–Pasta–Ulam (FPU) chain with the homogeneous fully nonlinear interaction potential which describes the propagation of acoustic wave in chains of touching beads without precompression. From the quasi-continuum approximation of the FPU chain, we first derive out a new type of wave equation which includes a second degree logarithmic nonlinear term. By finding an integrable factor equation, we obtain its Gaussian solitary wave solution. The result shows that if the effect of logarithmic nonlinearity can be balanced with the dispersion, the Gaussian solitary waves do exist for the second degree logarithmic wave equation in real physical models.
... Investigation of the exact solutions and dynamics of partial differential equations has been done by many researchers in several areas of sciences. Some methods have suggested for solutions of the partial differential equations such as tanh function method, Hirota bilinear method, exp-function method, G'/Gexpansion method, trial equation method, improved G'/G-expansion method, extended trial equation method, multiple extended trial equation method, Weierstrass elliptical function expansion method, Jacobian elliptical function method, first integral method, modified Kudryashov method, generalized Kudrayshov method and F-expansion method [1]- [3], [30,33,34,52,53,55,57,58,60], [15]- [17], [20]- [25], [39]- [50]. Ma and Fuchssteiner [38] proposed a powerful method to find the exact solutions of partial differential equations. ...
... In the recent years, this proposed method was improved by many researchers. Liu developed this approach called as a trial equation method and implemented to some evolution differential equations [32]- [34]. Recently this approach was further developed and called as extended trial equation method by Gurefe et al [21] and Pandir et al [46]. ...
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.
... In the domain of NLPDEs, the extended trail equation method [19] and generalized Arnous method [12] are very powerful tools that offer a systematic approach for finding soliton solutions from NLPDEs. In literature, many scientific problems have been solved using the trail equation method [14,[20][21][22]. ...
This study employs symbolic computational techniques to examine a third-order nonlinear Schrödinger equation as an extended model for soliton transmission in optical wave orientation. The extended trial equation method generates exact traveling wave solutions, including singular, soliton, rational function, and elliptic integral function solutions. In contrast, the generalized Arnous method constructs solutions such as dark-singular solitons, pure cubic dark solitons, and singular combinations, providing a diverse spectrum of solutions. The qualitative behavior of the resulting dynamical system is analyzed through bifurcation and chaotic analyses, revealing its sensitivity to initial condition variations. Graphical representations, including soliton profiles and phase portraits, illustrate the findings. All newly generated soliton solutions have not been published in the literature. These solutions are verified by substituting them back into the associated system using Maple Software. This study offers important insights into long-term behavior and helps design stable and robust systems across various fields, such as optical fibers, data transmission, nonlinear optics, and long-distance communication. These methods enhance understanding of nonlinear dynamical models, significantly contributing to studying complex systems in various scientific and engineering domains.
... By the trial equation method [25], Eq (2.13) has a polynomial solution Q(ζ), which satisfies the following equation: ...
This article studied the new traveling wave solutions of the cascaded model with higher-order dispersion effects combined with the effects of spatiotemporal dispersion and multiplicative white noise. In the process of exploring traveling wave solutions, a clever combination of the polynomial complete discriminant system was used to discover more forms of traveling wave solutions for this equation. In order to better observe and analyze the propagation characteristics of traveling wave solutions, we used Maple and Matlab software to provide two-dimensional and three-dimensional visualization displays of the equation solutions. Meanwhile, we also analyzed the internal mechanism of nonlinear partial differential equations using planar dynamical systems. The research results indicated that there are differences in the results of different forms of soliton solutions affected by external random factors, which provided more beneficial references for people to better understand the cascaded model with higher-order dispersion effects combined with the effects of spatiotemporal dispersion and multiplicative white noise, and helped people to more comprehensively understand the propagation characteristics of optical solitons. The solution method in this article was also applicable to the study of other nonlinear partial differential equations.
... In conversely, Liu (2005) has introduced an efficient trial equation method to derive the exact traveling wave solutions for nonlinear differential equations. Subsequently, several authors have introduced modified versions derived from Liu's method and applied them to solve a variety of nonlinear evolution equations (Liu 2006;Du 2010;Gurefe et al. 2013;Li and Wang 2019;Triki et al. 2019;Zhang et al. 2005a;Shang et al. 2023;Pandir and Ekin 2020;Hu et al. 2021;Triki and Wazwaz 2016). ...
In this paper, the dynamics of ultrashort light pulses inside an inhomogeneous optical medium that displays all orders of dispersion up to the fourth order as well as self-steepening, self-frequency shift, and linear gain or loss is investigated. By adopting a novel approach of the amplitude function derived from the trial equation method, three types of dipole solitons are discovered. Unlike conventional dipole solitons obtained in Kerr-type media, this new soliton structure is embedded within a continuous-wave (cw) background, which influences its propagation properties. Also, the dynamic behaviors of these new soliton pulses are thoroughly explored based on the modulation of medium and pulse parameters. The results show that the shape and dynamics of the dipole-pulses on a cw background can be controlled efficiently by choosing the gain or loss and the inter-modal dispersion functions appropriately. Besides, the study conducts numerical exploration into the collision dynamics involving two and three adjacent similar solitons impacted by the inhomogeneity of medium. Finally, the stability of the dipole soliton solution on a cw background is numerically studied under the finite perturbations.
... In this paper, we use the chirped transformation to convert Equation (1) into a two-dimensional planar dynamic system, which can represent its Hamiltonian properties, and find different equilibrium points and corresponding phase diagrams with respect to the parameters. Then we obtain all chirped solutions by using the complete discrimination system for polynomial method [33][34][35][36][37][38]. In real life, time-dependent perturbations are important. ...
In this paper, we focus on an extended nonlinear Schrödinger equation describing the pulse propagation in a nonlinear Schrödinger equation with self‐steepening and magneto‐optic effects. The existence of periodic and solitary solutions are proved based on the bifurcation method, and also, the Hamiltonian properties and the classification of its equilibrium points are obtained. The chirped solutions of the extended nonlinear Schrödinger equations are obtained by using the complete discrimination system for polynomial method, and under specific parameter conditions, three types of optical wave patterns are obtained to visualize the model. In particular, we consider the external perturbation terms to analyze the chaotic behavior of this equation.
... In this section the steps of the NVTEM are given to gain the wave solutions of a nonlinear partial differential equation (NPDE). This method is a developed version of the trial equation method which is first proposed by Liu (2005a;2005b Step1 We consider a NPDE as follows, P (u, u x , u y , u t , u xx , u xy , ...) = 0. ...
In this study a new version of the trial equation method (NVTEM) is employed to present the new exact wave solutions of the Fokas system which describes the propagation of nonlinear pulse in monomode optical fibers. Primarily this system is converted to a nonlinear ordinary differential equation (NODE) via the traveling wave transform to apply the proposed method. Various different exact wave solutions to the Fokas system are obtained such as rational function, exponential function, hyperbolic function and Jacobi elliptic function solutions. The exhibition of the solutions are given by 3D plots together with the corresponding 2D plots. The outcomes have shown that the proposed technique is abundant to achieve different wave solutions of many nonlinear differential equations in the field of optics.
... In the recent past, Liu defined the trial equation method in that the elliptic differential equations and the complete discrimination system for polynomials are used [17]. Then, a new version of the trial equation method for the nonlinear problems with rank inhomogeneous is introduced by Liu [18]. ...
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.
... However, the type of their solution is only the solitary wave solution. But in this paper, we investigate exact chirped solutions of the perturbed resonant NLSE with dual-power law nonlinearity by means of the complete discrimination system for polynomial method (Liu 2010(Liu , 2004(Liu , 2005a(Liu , 2006a(Liu , 2007(Liu , 2008(Liu , 2005bKai and Yin 2022;Kai et al. 2020;Kai and Chen 2021;Chen et al. 2002;Wei et al. 2023;Wang 2022) and the trial equation method (Liu 2006b(Liu , 2005c(Liu , 2011Hu et al. 2021). Due to the complicated form of the chirped wave transformation, chirped solutions are more laborious to acquire than envelope traveling wave solutions in mathematics. ...
In this paper, the perturbed resonant nonlinear Schrödinger equation with dual-power law nonlinearity describing the pulse phenomena in nonlinear optics is investigated. By utilizing the complete discrimination system for polynomial method and the trial equation method, a variety of exact solutions of this equation have been acquired, including rational solutions, triangular function periodic solutions, solitary wave solutions, elliptic function double periodic solutions, as well as the corresponding nonlinear chirps. Besides, several representative two-dimensional graphs are plotted at specific parameters, which can supply a more comprehensive physical interpretation of the complex nonlinear model based on the intuitive identification of the morphology of optical waves.
... By introducing the following trial equation 39,40 ðÞ ...
The nonlinear coupled Schrödinger equation in fiber Bragg gratings is studied in this paper. The existence of soliton solutions and periodic solutions are proved by qualitative analysis, and exact solutions are given, as well as the parameter condition of each solution is described. Then the modulation instability (MI) analysis is carried out and the linear stability criterion is given. In particular, external perturbation terms are introduced to prove that the equation exists chaotic behaviors.
... This idea was developed in the following years by many scientists. Liu called this approach the trial equation method and applied it to some nonlinear evolution equations [37][38][39]. Recently this approach was subsequently improved and named as extended trial equation method by Pandir et al [16,17]. ...
New solitary wave solutions for the Korteweg-de Vries (KdV) equation by a new version of the trial equation method are attained. Proper transformation reduces the Korteweg-de Vries (KdV) equation to a quadratic ordinary differential equation that is fully integrated using the new version trial equation approach. The family of solitary wave solutions of the reduced equation ensures a combined expression for the Korteweg-de Vries (KdV) equation, which contains exact solutions derived in recent years using different integration methods. The analytic solution of the reduced equation permits to find exact solutions for the Korteweg-de Vries (KdV) equation, providing a variety of new solitary wave solutions that have not been reported before.
... Compared with other methods, the trial equation method is more straightforward, practical, and all-encompassing. Later, he used it to solve a significant number of equations [55][56][57][58][59][60][61][62][63] that are famous in many fields, such as Sine-Gördon equation, NNV equation, and a class of generalized Ginzburg-Landau equation. Then, many scholars introduced modified versions based on Liu's method. ...
The coupled Schrödinger-Korteweg-de Vries equation is a critical system of in nonlinear evolution equations. It describes various processes in dusty plasma, such as Langmuir waves, dust-acoustic waves, and electromagnetic waves. This paper uses the generalized coupled trial equation method to solve the equation. By the complete discrimination system for polynomial, a series of exact traveling wave solutions are obtained, including discontinuous periodic solutions, solitary wave solutions, and Jacobian elliptical function solutions. In addition, to determine the existence of the solutions and understand their properties, we draw three-dimensional images of the modules of the solutions with Mathematica. We obtain more comprehensive and accurate solutions than previous studies, and the results give the system more profound physical significance.
... We can classify the roots of F(G) [16][17][18][19][20][21][22][23][24] and solve the integral (34). ...
The paper addresses optical solitons in magneto-optic waveguides that are studied using Kudryashov’s law of nonlinear refractive index in the presence of chromatic dispersion and Hamiltonian-type perturbation terms. The trial solution approach yielded a variety of soliton solutions, which are listed in this paper.
... Based on this thought, various methods have been developed and applied to investigate the exact or approximate solutions of nonlinear mathematical models. Some of those methods are, respectively, the trial equation method [1], the Kudryashov method [2], the first integral method [3], the extended trial equation method [4], the new extended direct algebraic method [5], the modified exponential function method [6], the sine-cosine method [7], the extended tanh-function method [8], the variational iteration method [9], Adomian decomposition method [10], the Chebyshey-Tau method [11]. Apart from these, a new method called the "new function method" has been proposed in the paper [12]. ...
In this article, the new function method is used to obtain the wave solutions of the nonlinear Klein-Gordon equation. Since the Klein-Gordon equation is a nonlinear partial differential equation containing exponential functions, it was decided to apply the new function method, which was defined on the assumption of a nonlinear auxiliary differential equation containing exponential functions. Thus, it aims to reach wave solutions not found in the literature. The considered method can be easily applied to this type of nonlinear problem that is difficult to solve and gives us solutions. Here, two new exact solutions are obtained. Then two and three-dimensional density and contour graphs are drawn by selecting the appropriate parameters to analyze the physical behavior of these solutions. The Mathematica package program was effectively used in all calculations and graphic drawings.
... Because the obtained findings allow the analysis of the event, for this reason, there are various methods related to numerical or exact solutions of partial differential equations in the literature. Some of them are respectively the trial equation method [1], the extended trial equation method [2], the newfunction method [3][4][5][6][7], the improved Bernoulli subequation function method [8][9][10], Kudryashov method [11,12], the sine-Gordon equation expansion method [13][14][15][16], generalized auxiliary equation method [17], first integral method [18], new extended direct algebraic method [19], Hirota bilinear method and the tanh-coth method [20,21], the modified exponential function method [22], and Chebyshev-Tau method [23]. ...
In this study, the traveling wave solutions of a nonlinear partial differential equation (NPDE) called as Geophysics Korteweg-de Vries (GpKdV) equation are obtained using the modified exponential function method (MEFM). Coriolis effect is stated with the help of this model used in geophysics. The nonlinear model has a Coriolis coefficient representing this effect.
... In recent years, the scholars have not slowed down their pace in finding different menthods to present the solitary wave solutions in some fields of physical science [1]. Some scholars have put forward effective methods to solve these problems, such as Ma's and Li's changed objective capacity technique [2], Liu's canonical transformation method [3] and trial equation method [4][5][6][7][8][9], Ma's and Zhu's multiple empirical function method [10] and other direct expansion methods [11][12][13][14][15]. Moreover, the exact solutions for some nonlinear differential equations are classifiled by using the complete discrimination system for polynomials [16][17][18][19][20][21][22][23][24]. ...
In this paper, the traditional DSW conditions was presented. The nonlinear incomplete differential conditions were converted into common differential conditions through travelling wave transformation, and all the definite voyaging wave arrangements for the traditional DSW conditions can characterized by use a complete separation framework for polynomials. The specific voyaging wave arrangements existed as periodic solutions of geometrical functions and sane numbers, lone wave arrangements, doubly occasional arrangements of elliptic functions.
... In scientific studies in the literature, there are various methods to investigate the solutions of such equations. Some of these methods in the literature; generalized tanh function method [1], the modified extended tanh-function method [2][3], the generalized Bernoulli sub-equation function method [4][5][6], The trial equation method [7][8][9][10][11], the first integral method [12], the modified exponential function method [13][14][15][16][17] and many more methods. ...
In this study, wave solutions of the (3+1) dimensional Jimbo-Miwa equation and two different phenomena of the solution, fusion and fission, are obtained using the modified exponential functionmethod. In order to get more possible solutions, two different cases are investigated due to the nature of the modified exponential function method. When the resulting solutions are analyzed, trigonometric, hyperbolic and rational functions are obtained. It was checked whether the solution functions found by the Wolfram Mathematica software provided the (3+1) dimensional potential Jimbo-Miwa equation. Two and three dimensional graphs, contour and density graphs of the solution function were get by determining the appropriate parameters.
... To solve Eq. (1), we first use the method of separation of variables and the traveling wave transformation to transform it into the ordinary differential equation; subsequently, we use the trial equation method [16] to rewrite it into the integral form. After yielding its integral form, the complete discrimination system for polynomial method [17][18][19][20][21][22] is applied to this equation. Qualitative properties like phase portraits are obtained, and the exact solutions, including the elliptic function and soliton solutions, are also obtained. ...
This paper considers a fourth-order time-fractional partial differential equation with Riemann–Liouville definition. We first use the general method of separation of variables to transform the original equation into an ordinary differential equation and subsequently apply the trial equation method to obtain its integral form. The complete discrimination system for polynomial method(CDSPM) is also adopted herein. By applying this method, dynamic properties such as phase portraits are determined. The results suggest that the soliton solution coexists with the periodic solution as long as the homoclinic orbits exist. Moreover, to directly show our conclusions, the corresponding exact solutions to this equation are presented using this method.
In this paper, the nonlinear Schrödinger equation combining quadratic-cubic nonlinearity is considered, which can be represented by an approximate model of relatively dense quasi-one-dimensional Bose-Einstein condensate. Based on the bifurcation theory, we proved the existence of solitary and periodic solutions. The methods we take are the trial equation method and the complete discrimination system for polynomial method. Therefore, we obtain the exact chirped solutions, which are more abundant in type and quantity than the existing results, so that the equation has more profound physical significance. These two methods are rigorously mathematical derivation and calculations, rather than based on certain conditional assumptions. In addition, we give some specific parameters to graphing the motion of the solutions, which helps to understand the propagation of nonlinear waves in fiber optic systems.
In this paper, we study the higher-order nonlinear Schrödinger equation of surface gravity wave evolution at the critical point . We use dynamic systems to analyse the types of solutions to this equation. Nineteen analytic chirped solutions are obtained by using the chirped wave transformation and polynomial complete discriminant system method. Finally, we add different disturbance terms to the equation and observe chaotic behaviour in the system. These results show that under different parameter conditions the frequency of the surface gravity wave changes as the amplitude changes. Therefore, the chirp shows a variant of patterns such as singular and periodic structures.
Based on the nonlinear Schrödinger equation, we investigate the exact nonlinear chirps of the dynamic model for the propagation of optical solitons in nonlinear metamaterials. The trial equation method and the complete discrimination system for polynomial method are utilized to construct dark soliton, bright soliton, kink soliton, anti-kink soliton, Jacobi elliptic function solutions, as well as singular rational and singular biperiodic solutions. Notably, we have derived new solutions in the forms of the kink soliton, anti-kink soliton, and Jacobi elliptic function. Therefore, we provide a comprehensive categorization of all solutions, along with the parameter requirements for determining their existence. Furthermore, we also receive corresponding amplitude and chirp graphs for specific physical parameters, which aid in understanding the propagation characteristics of chirped solutions in optical metamaterials.
In this study, the dissipation problem of nonlinear pulse in mono mode optical fibers which is governed by the Fokas system (FS) is considered. The solutions of this system have an important role in comprehending the different wave structures in physical settings. Therefore, a new version of the trial equation method (NVTEM) is employed to present the new exact wave solutions of the FS. The advantage of the NVTEM is to use different root possibilities of a polynomial which shape the solutions of the related model. Primarily this system is converted to a nonlinear ordinary differential equation (NODE) via the traveling wave transform to apply the proposed method. Various exact wave solutions to the FS are obtained such as rational function, exponential function, hyperbolic function, and Jacobi elliptic function solutions. Thus, we have revealed solutions featly which are unlike the wave solutions previously found by other analytical methods. The present results depict the formation and development of such waves and their interactions. The exhibition of the solutions is given by 3D plots together with the corresponding 2D plots. The outcomes have shown that the proposed technique is abundant in achieving different wave solutions of many nonlinear differential equations in the field of optics.
The paper investigates the coupled system of NLSE for the generalized Kudryashov’s equation in magneto-optic waveguides, in which the trial equation method and the complete discriminant system for a polynomial are employed to obtain a rich set of exact solutions, including rational solutions, solitary wave solutions, triangular function solutions, and elliptic function solutions. Numerical simulations are also conducted on the solutions under specific parameters to visually depict the structure and characteristics.
In this work, traveling wave solutions of (1+1)-dimensional Landau-Ginzburg-Higgs and Duffing nonlinear partial differential equations, which are examples of mathematical modeling, are obtained and analyzed using the modified exponential function method. In order to facilitate the physical interpretation of the mathematical models represented by these equations, simulations of the behavior of the mathematical model as three-dimensional, contour, density and two-dimensional graphics are given using a package program with the help of appropriate parameters. It has been shown that the modified exponential function method effectively investigates the solutions of (1+1)-dimensional Landau-Ginzburg-Higgs and Duffing equations.
In this paper, the perturbed Gerdjikov–Ivanov equation with spatio-temporal dispersion is investigated. The trial equation method, the complex envelope travelling wave transformation, and the complete discriminant system for polynomial method, respectively, are all utilized. As a consequence, twenty-eight exact chirped solutions are obtained, and some representative two-dimensional patterns under concrete parameters are presented. The results acquired are conducive to studying the dynamic behavior of solitons.
This paper aims to transform the nonlinear development equation into elementary integral form by the trial equation method. The Gardner equation which can be used to describe stratified fluids is studied. By applying the trial equation method, 47 types of traveling wave solutions are obtained. The method is applied in the generalized Gardner equation with variable coefficients. It shows that the method is very effective for solving some linear or simple nonlinear partial differential equations with variable coefficients.KeywordsTrial equation methodGardner equationExact solutionVariable coefficientTraveling wave solution
It is of great significance to explore the physical properties of optical propagation, thus the scholars are keen on the physical essence represented by classical mathematical physics equations. In this paper, a higher order nonlinear Schrödinger equation describing the behavior of polarization mode in optical fibers is firstly analyzed qualitatively, and the existence of periodic and soliton solutions is proved by using bifurcation method. Secondly, all chirped wave patterns with special form for the equation are obtained by using the complete discrimination system for polynomial method and direct integral method, the chirped rational functions and some elliptic functions wave patterns are initially found. As a result, the parameter stability of these patterns is given for the first time, which shows the variant of patterns as the parameters change, and the graphs of several typical patterns are drawn.
This paper investigates the modified cubic–quintic complex Ginzburg–Landau equation with parabolic law that depictures the propagation of chirped wave. The used methodologies are the trial equation method and the complete discriminant system for polynomial method. As a result, nineteen exact chirped solutions are obtained, and several typical numerical simulations under specific parameter conditions are given.
In this paper, we study the resonant nonlinear Schrödinger equation which describes the propagation of optical solitons in optical fibers. Through the trial equation method and the complete discrimination system for polynomial we obtain abundant optical wave propagation patterns for the model and divide them into three categories. Besides, we analyze the topological stability of these patterns. Ultimately, we give physical representations of optical wave patterns and draw some figures to show their temporal-space structures.
In this paper, we study the optical solitons for Kudryashov’s law with dual form of generalized non-local nonlinearity in optical fibers. Through the trial equation method and the complete discrimination system for polynomial, we get various optical wave patterns for the model. Among those, the new patterns including rational function pattern, trigonometric function pattern and elliptic function pattern. In addition, we draw some graphs to show the propagation of optical wave patterns and the temporal-space structures.
Under investigation in this paper is a high-order nonlinear Schrödinger equation with weak non-local nonlinearity. The complete discrimination system for polynomial method is applied to conduct qualitative analysis, and carry out the existence of periodic and soliton solutions. We construct concrete solutions to verify our conclusion and some new optical traveling wave solutions have also been found. Furthermore, the parameters condition of existence of each solution is presented clearly.
What the paper will show is how we attempt to study optical solitons of the nonlinear Schrödinger (NLS) equation with anti-cubic (AC) nonlinearity which describes the propagation of pulses in optical metamaterials. By using trial equation method and the complete discrimination system for polynomial, we obtain all the possible forms of optical propagation patterns for the model. Finally, we draw the graphs for several typical patterns and show their temporal-space structures.
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.
The propagation of optical solitons in optical fibers with the generalized Kudryashov’s refractive index is described by a high order nonlinear Schrödinger equation. The main feature is that the amplitude and width of the pulse can be changed with arbitrary power. By trial equation method, a series of propagation patterns of optical waves are obtained. In addition, the graphs of two-dimensional and three-dimensional solutions are illustrated, and then the existence of all those patterns of optical waves is proved.
In this paper, according to the complete discriminant system for polynomial method, a series of analytical solutions of the perturbed nonlinear Schrödinger equation with variable coefficient and Kerr law are obtained. These solutions show the abundant propagation patterns of optical waves with Kerr law.
Optical propagation in anti-cubic nonlinear optical metamaterials is investigated by solving the higher order nonlinear Schrödinger equation. A unified integral approach is applied to the governing equation. A complete list of exact envelope patterns is obtained to show the richness of the propagation patterns, which include solitons, singular and quasi-periodic patterns and double periodic patterns. In practice, by adjusting or controlling the physical parameters, the needed pattern can be obtained.
This paper studies the traveling wave modes for nonlinear Biswas–Milovic equation in magneto-optical wave guide coupling system with Kudryashov’s law of refractive index. By the complete discrimination system for polynomial and trial equation method, we derive a series of traveling wave solutions of the coupling system. These results show the abundant propagation patterns. In particular, by analyzing the topological stability and dynamic behavior we get four stable modes of this system. Under some special parameters, we give the concrete representations of solutions.
In a previous paper (Lou S-y 1995 J. Phys. A: Math. Gen. 28 7227), a generalized dromion structure was revealed for the (2+1)-dimensional KdV equation, which was first derived by Boiti et al (Boiti M, Leon J J P, Manna M and Pempinelli F 1986 Inverse Problems 2 271) using the idea of the weak Lax pair. In this paper, using a Bäcklund transformation and the variable separation approach, we find there exist much more abundant localized structures for the (2+1)-dimensional KdV equation. The abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution. Some special types of dromion solution, lumps, breathers, instantons and the ring type of soliton, are discussed by selecting the arbitrary functions appropriately. The dromion solutions can be driven by sets of straight-line and curved-line ghost solitons. The dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves. The breathers may breathe both in amplitude and in shape.
Using a complete discrimination system for polynomials and
elementary integral method, we obtain the travelling solutions for
triple sine–Gordon equation. This method can be applied to similar
problems and has general meaning.
The solitary wave solutions of two types of variant Boussinesq equations are obtained by using a homogeneous balance method.
The solitary wave solutions of the approximate equations for long water waves, the coupled KdV equations and the dispersive long wave equations in 2 + 1 dimensions are constructed by using a homogeneous balance method.
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