Abundant Traveling Wave Solutions Research Articles

An investigation of abundant traveling wave solutions of complex nonlinear evolution equations: The perturbed nonlinear Schrodinger equation and the cubic-quintic Ginzburg-Landau equation

AbstractIn this article, the two variables (G′/G,1/G)-expansion method is suggested to obtain abundant closed form wave solutions to the perturbed nonlinear Schrodinger equation and the cubic-quintic Ginzburg-Landau equation arising in the analysis of various problems in mathematical physics. The wave solutions are expressed in terms of hyperbolic function, the trigonometric function, and the rational functions. The method can be considered as the generalization of the familiar (G′/G)-expansion method established by Wang et al. The approach of this method is simple, standard, and computerized. It is also powerful, reliable, and effective.

Symbolic computation and abundant travelling wave solutions to KdV–mKdV equation

In this article, the novel (G ′/G)-expansion method is successfully applied to construct the abundant travelling wave solutions to the KdV–mKdV equation with the aid of symbolic computation. This equation is one of the most popular equation in soliton physics and appear in many practical scenarios like thermal pulse, wave propagation of bound particle, etc. The method is reliable and useful, and gives more general exact travelling wave solutions than the existing methods. The solutions obtained are in the form of hyperbolic, trigonometric and rational functions including solitary, singular and periodic solutions which have many potential applications in physical science and engineering. Many of these solutions are new and some have already been constructed. Additionally, the constraint conditions, for the existence of the solutions are also listed.

Traveling wave solutions of generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony and simplified modified form of Camassa–Holm equation exp(–φ(η)) – Expansion method

In this article, we established abundant traveling wave solutions for nonlinear evolution equations. The exp(–φ(η))-expansion method is used to construct traveling wave solutions for the generalized Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation and Simplified Modified form of Camassa–Holm equation. The traveling wave solutions are expressed in terms of the hyperbolic functions, the trigonometric functions and the rational functions. The proposed solutions are found to be important for the explanation of some practical physical problems in mathematical physics and engineering.

An ansatz for solving nonlinear partial differential equations in mathematical physics.

In this article, we introduce an ansatz involving exact traveling wave solutions to nonlinear partial differential equations. To obtain wave solutions using direct method, the choice of an appropriate ansatz is of great importance. We apply this ansatz to examine new and further general traveling wave solutions to the (1+1)-dimensional modified Benjamin–Bona–Mahony equation. Abundant traveling wave solutions are derived including solitons, singular solitons, periodic solutions and general solitary wave solutions. The solutions emphasize the nobility of this ansatz in providing distinct solutions to various tangible phenomena in nonlinear science and engineering. The ansatz could be more efficient tool to deal with higher dimensional nonlinear evolution equations which frequently arise in many real world physical problems.

A Novel $$\left( {{G}'/G} \right) $$ G ′ / G -Expansion Method and its Application to the (3 + 1)-Dimensional Burger’s Equations

In this article, a novel $$\left( {{G}'/G} \right) $$ -expansion method is used to look for the traveling wave solutions of nonlinear evolution equations. We construct abundant traveling wave solutions involving parameters to the (3 + 1)-dimensional integrable Burger’s equations by means of the suggested method. The performance of the method is reliable, useful and gives more new general exact solutions than the existing methods. The novel $$\left( {{G}'/G} \right) $$ -expansion method provides more general forms of solutions.

A Note on Novel ( G'/G) -expansion Method in Nonlinear Physics

Abstract: The exact solutions of nonlinear evolution equations (NLEEs) play a critical role to make known the internal mechanism of complex physical phenomena. In this article, we construct the traveling wave solutions of the (1+1)-dimensional KdV equation and the Banjamin-Ono equation by means of the novel (G0/G)-expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, trigonometric and rational functions. It is shown that the novel (G0/G)-expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations.

New (G′/G)-expansion method and its application to the Zakharov-Kuznetsov–Benjamin-Bona-Mahony (ZK–BBM) equation

In this article, new (G′/G)-expansion method is used to look for the traveling wave solutions of nonlinear evolution equations and abundant traveling wave solutions to the Zakharov-Kuznetsov–Benjamin-Bona-Mahony equation are constructed. The performance of the method is reliable, useful and gives more new general exact solutions than the existing methods.

Traveling wave solutions of the Boussinesq equation via the new approach of generalized (G′/G)-expansion method

Exact solutions of nonlinear evolution equations (NLEEs) play a vital role to reveal the internal mechanism of complex physical phenomena. In this work, the exact traveling wave solutions of the Boussinesq equation is studied by using the new generalized (G′/G)-expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, trigonometric, and rational functions. It is shown that the new approach of generalized (G′/G)-expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations in mathematical physics and engineering.PACS05.45.Yv, 02.30.Jr, 02.30.Ik

A novel (G′/G)-expansion method and its application to the Boussinesq equation

In this article, a novel (G′/G)-expansion method is proposed to search for the traveling wave solutions of nonlinear evolution equations. We construct abundant traveling wave solutions involving parameters to the Boussinesq equation by means of the suggested method. The performance of the method is reliable and useful, and gives more general exact solutions than the existing methods. The new (G′/G)-expansion method provides not only more general forms of solutions but also cuspon, peakon, soliton, and periodic waves.

Application of the new approach of generalized (G' /G) -expansion method to find exact solutions of nonlinear PDEs in mathematical physics

The exact solutions of nonlinear evolution equations (NLEEs) play a crucial role to make known the internal mechanism of complex physical phenomena. In this article, we construct the traveling wave solutions of the Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZK-BBM) equation by means of the new approach of generalized (G′ /G) -expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, trigonometric, and rational functions. It is shown that the new approach of generalized (G′ /G) -expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations. BIBECHANA 10 (2014) 58-70 DOI: http://dx.doi.org/10.3126/bibechana.v10i0.9312

Traveling Wave Solutions of Nonlinear Evolution Equations Via the New Generalized (G′/G)-Expansion Method

The exact solutions of nonlinear evolution equations (NLEEs) play a crucial role to make known the internal mechanism of compound physical phenomena. In this article, we implement the new generalized (G’/G)-expansion method for seeking the exact solutions of NLEEs via the Benjamin-Ono equation and achieve exact solutions involving parameters. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, trigonometric, and rational functions. It is shown that the new approach of generalized G’/G -expansion method is a powerful and concise mathematical tool for solving nonlinear evolution equations.

Solitary wave solutions of fifth-order (1+1)-dimensional Caudrey-Dodd-Gibbon equation

The manuscript deals with the abundant travelling wave solutions of the Caudrey-Dodd-Gibbon (CDG) equation which have been obtained in a uniform way by using alternative (G`/G)–expansion method wherein the generalized Riccati equation is used. Moreover, a relatively new technique which is called (U`/U)-expansion is also used to find solitary wave solutions of CDG equation. The solutions obtained in this article may be imperative and significant for the explanation of some practical physical phenomena. Numerical results coupled with the graphical representation explicitly reveal the complete reliability and high efficiency of the proposed algorithms. Key words: (G`/G)-expansion method, travelling wave solutions, (U`/U)-expansion method, Caudrey-Dodd-Gibbon (CDG) equation, nonlinear evolution equations.

Traveling wave solutions of nonlinear evolution equations via the enhanced (G′/G)-expansion method

Abstract In this article, an enhanced ( G ′/ G )-expansion method is suggested to find the traveling wave solutions for the modified Korteweg de-Vries (mKDV) equation. Abundant traveling wave solutions are derived, which are expressed by the hyperbolic and trigonometric functions involving several parameters. The efficiency of this method for finding these exact solutions has been demonstrated. It is shown that the proposed method is effective and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics.

Extended generalized Riccati equation mapping method for the fifth-order Sawada-Kotera equation

In this article, the generalized Riccati equation mapping together with the basic (G′/G)-expansion method is implemented which is advance mathematical tool to investigate nonlinear partial differential equations. Moreover, the auxiliary equation G′(ϕ) = h + f G(ϕ) + g G2(ϕ) is used with arbitrary constant coefficients and called the generalized Riccati equation. By applying this method, we have constructed abundant traveling wave solutions in a uniform way for the Sawada-Kotera equation. The obtained solutions of this equation have vital and noteworthy explanations for some practical physical phenomena.

An Alternative (G’/G) -Expansion Method with Generalized Ric-cati Equation: Application to Fifth Order (1+1)-Dimensional Kaup–Keperschmidt Equation

In this paper, abundant traveling wave solutions of the Kaup-Keperschmidt (KK) equation have been obtained in a uniform way by using an alternative (G'/G)-expansion method wherein the generalized Riccati equation is used. The obtained solutions in this work are imperative and significant for the explanation of some practical physical phenomena. It is shown that the alternative (G'/G)-expansion method together with the generalized Riccati equation provides advance mathematical tool for solving nonlinear partial differential equations. Numerical results coupled with the graphical representation explicitly reveal the complete reliability and high efficiency of the proposed algorithm.

Abundant traveling wave solutions of the compound KdV-Burgers equation via the improved (G′/G)-expansion method

In this article, we investigate the compound KdV-Burgers equation involving parameters by applying the improved (G′/G)-expansion method for constructing some new exact traveling wave solutions including solitons and periodic solutions. The second order linear ordinary differential equation with constant coefficients is used, in this method. The obtained solutions are presented through the hyperbolic, the trigonometric and the rational functions. Further, it is significant to point out that some of our solutions are in good agreement for special cases with the existing results which validates our other solutions. Moreover, some of the obtained solutions are described in the figures.

Exact travelling wave solutions of nonlinear evolution equations by using the [formula omitted]-expansion method

In this work, we established abundant travelling wave solutions for some nonlinear evolution equations. The G ′ G -expansion method was used to construct travelling wave solutions of nonlinear evolution equations. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. This method presents a wider applicability for handling nonlinear wave equations.

Towered waves and anti-waves in the generalized Degasperis–Procesi equation

New traveling wave solutions of the generalized Degasperis–Procesi equation are investigated. The solutions are characterized by three parameters. Using an improved qualitative method, abundant traveling wave solutions, such as smooth waves, peaked waves, cusped waves, compacted waves, looped waves and fractal-like waves, are obtained. Especially, some strange composite wave solutions such as towered waves and their anti-waves are first given. We also study the limiting behavior of all periodic solutions as the parameters trend to some special values.

The (G′/G)‐Expansion Method for Abundant Traveling Wave Solutions of Caudrey‐Dodd‐Gibbon Equation

We construct the traveling wave solutions of the fifth‐order Caudrey‐Dodd‐Gibbon (CDG) equation by the (G′/G)‐expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, the trigonometric, and the rational functions. It is shown that the (G′/G)‐expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations.

The (G′/G, 1/G)-expansion method and its application to travelling wave solutions of the Zakharov equations

The (G'/G, 1/G)-expansion method for finding exact travelling wave solutions of nonlinear evolution equations, which can be thought of as an extension of the (G'/G)-expansion method proposed recently, is presented. By using this method abundant travelling wave solutions with arbitrary parameters of the Zakharov equations are successfully obtained. When the parameters are replaced by special values, the well-known solitary wave solutions of the equations are rediscovered from the travelling waves.