Solutions Of Nonlinear Evolution Equations Research Articles

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.

New generalized and improved (G′/G)-expansion method for nonlinear evolution equations in mathematical physics

Abstract In this article, new extension of the generalized and improved ( G ′/ G )-expansion method is proposed for constructing more general and a rich class of new exact traveling wave solutions of nonlinear evolution equations. To demonstrate the novelty and motivation of the proposed method, we implement it to the Korteweg-de Vries (KdV) equation. The new method is oriented toward the ease of utilize and capability of computer algebraic system and provides a more systematic, convenient handling of the solution process of nonlinear equations. Further, obtained solutions disclose a wider range of applicability for handling a large variety of nonlinear partial differential equations.

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.

Exact traveling wave solutions of the KP-BBM equation by using the new approach of generalized (G′/G)-expansion method

The new approach of the generalized (G′/G)-expansion method is an effective and powerful mathematical tool in finding exact traveling wave solutions of nonlinear evolution equations (NLEEs) in science, engineering and mathematical physics. In this article, the new approach of the generalized (G′/G)-expansion method is applied to construct traveling wave solutions of the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation. The solutions are expressed in terms of the hyperbolic functions, the trigonometric functions and the rational functions. By means of this scheme, we found some new traveling wave solutions of the above mentioned equation.

English

The functional variable method is a powerful mathematical tool for obtaining exact solutions of nonlinear evolution equations in mathematical physics. In this paper, the functional variable method is used to establish exact solutions of the (2+1)-dimensional Kadomtsov-Petviashivilli-Benjamin-Bona-Mahony (KP-BBM) equation, the (2+1)-dimensional Konopelchenko-Dubrovsky equation, the (3+1)-dimensional Burgers equation and the (3+1)- dimensional Jimbo-Miwa equation. The exact solutions of these four nonlinear equations including solitary wave solutions and periodic wave solutions are obtained. It is shown that the proposed method is effective and can be applied to many other nonlinear evolution equations. Comparison between our results obtained in this paper and the well-known results obtained by different authors using different methods are presented. Key words: Functional variable method, nonlinear evolution equations, exact solutions, solitary wave solutions, Periodic wave solutions.

A New Extended (G'/G)-Expansion Method to Find Exact Traveling Wave Solutions of Nonlinear Evolution Equations

In this paper, we propose a new extended (G ' =G)-expansion method to construct exact traveling wave solutions for nonlinear evolution equations. To check the validity and effectiveness of our method, we imple- ment it to the (2+1)-dimensional typical breaking soliton equation.The results that we get are more general and successfully recover most of the previously known solutions which have been found by other sophisticated methods. Many of these solution are found for the first time. Moreover, our method is direct, concise, elementary, effective and can be used for many other nonlinear evolution equations.

Study of Nonlinear Evolution Equations to Construct Traveling Wave Solutions via the New Approach of the Generalized (G'/G) -Expansion Method

Exact solutions of nonlinear evolution equations (NLEEs) play very important role to make known the inner mechanism of compound physical phenomena. In this paper, the new generalized (G'/G)-expansion method is used for constructing the new exact traveling wave solutions for some nonlinear evolution equations arising in mathematical physics namely, the (3+1)-dimensional Zakharov-Kuznetsov equation and the Burgers equation. As a result, the traveling wave solutions are expressed in terms of hyperbolic, trigonometric and rational functions. This method is very easy, direct, concise and simple to implement as compared with other existing methods. This method presents a wider applicability for handling nonlinear wave equations. Moreover, this procedure reduces the large volume of calculations.

Topological soliton solutions for some nonlinear evolution equations

In this paper, the topological soliton solutions of nonlinear evolution equations are obtained by the solitary wave ansatz method. Under some parameter conditions, exact solitary wave solutions are obtained. Note that it is always useful and desirable to construct exact solutions especially soliton-type (dark, bright, kink, anti-kink, etc.) envelope for the understanding of most nonlinear physical phenomena.

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.

Traveling wave solutions of the (2 + 1)-dimensional Zoomeron equation and the Burgers equations via the MSE method and the Exp-function method

The modified simple equation (MSE) method is promising for finding exact traveling wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics. In this letter, we investigate solutions of the (2+1)-dimensional Zoomeron equation and the (2+1)-dimensional Burgers equation by using the MSE method and the Exp-function method. The competence of the methods for constructing exact solutions has been established.

Almost periodic viscosity solutions of nonlinear evolution equations in Carnot groups

The aim of this paper is to establish existence and uniqueness of time almost periodic viscosity solutions to first-order evolution equations in Carnot groups.

Functional variable method to study nonlinear evolution equations

Abstract The functional variable method is a powerful solution method for obtaining exact solutions of nonlinear evolution equations. This method presents a wider applicability for handling nonlinear wave equations. In this paper, the functional variable method is used to construct exact solutions of Davey-Stewartson equation, generalized Zakharov equation, K(m, n) equation with generalized evolution term, (2 + 1)-dimensional long-wave-short-wave resonance interaction equation and nonlinear Schrödinger equation with power law nonlinearity. The obtained solutions include solitary wave solutions, periodic wave solutions.

(G/G, 1/G)-expansion method for traveling wave solutions of (2+1) dimensional generalized KdV, Sin Gordon and Landau-Ginzburg-Higgs Equations

The-expansion method is one of the most direct and effective method for obtaining exact solutions of nonlinear partial differential equations (PDEs). In the present article, we construct the exact travelling wave solutions of nonlinear evolution equations in mathematical physics via (2+1) dimensional generalized KdV, Sin Gordon Equation and Landau-Ginzburg-Higgs Equation by -expansion method, where satisfies the auxiliary ordinary differential equation (ODE) where and are arbitrary constants. Key words: (2+1) dimensional generalized KdV, Sin Gordon Equation, Landau- Ginzburg-Higgs equation, -expansion method, auxiliary equation, travelling wave solutions.

Assessment of the further improved (G'/G)-expansion method and the extended tanh-method in probing exact solutions of nonlinear PDEs

PACS numbers02.30.Jr, 05.45.Yv, 02.30.Ik

Exact solutions of modified Zakharov–Kuznetsov equation by the homogeneous balance method

In this article, the homogeneous balance method is used to construct exact traveling wave solutions for the modified Zakharov–Kuznetsov equation using the Riccati equation. This method is a powerful tool for obtaining exact traveling wave solutions of nonlinear evolution equation. Many exact traveling wave solutions of nonlinear modified Zakharov–Kuznetsov equation are constructed which contain soliton and periodic solutions. The results reveal that the method is very effective and simple.

Exact solutions of the (2+1)-dimensional cubic Klein–Gordon equation and the (3+1)-dimensional Zakharov–Kuznetsov equation using the modified simple equation method

Exact solutions of nonlinear evolution equations (NLEEs) play a vital role to reveal the internal mechanism of complex physical phenomena. In this article, we implemented the modified simple equation (MSE) method for finding the exact solutions of NLEEs via the (2+1)-dimensional cubic Klein–Gordon (cKG) equation and the (3+1)-dimensional Zakharov–Kuznetsov (ZK) equation and achieve exact solutions involving parameters. When the parameters are assigned special values, solitary wave solutions are originated from the exact solutions. It is established that the MSE method offers a further influential mathematical tool for constructing exact solutions of NLEEs in mathematical physics.

Solitary-wave solutions of the Klein-Gordon equation with quintic nonlinearity

In this paper, the (G′/G)-expansion method is used to obtain exact solitary-wave and periodic-wave solutions for nonlinear evolution equations arising in mathematical physics with the aid of symbolic computations, namely, the Klein-Gordon equation with quintic nonlinearity. Our work is motivated by the fact that the (G′/G)-expansion method provides not only more general forms of solutions, but also periodic and solitary waves. As a result, hyperbolic function solutions and trigonometric function solutions with parameters are obtained. The method is straightforward and concise, and its application is promising for other nonlinear evolution equations in mathematical physics.

The averaging principle and periodic solutions for nonlinear evolution equations at resonance

We study the existence of T-periodic solutions (T>0) for the first order differential equations being at resonance at infinity, where the right hand side is the perturbations of a sectorial operator. Our aim is to prove an index formula expressing the topological degree of the associated translation along trajectories operator on appropriately large ball, in terms of special geometrical assumptions imposed on the nonlinearity. We also prove that the geometrical assumptions are generalizations of well known Landesman–Lazer and strong resonance conditions. The obtained index formula is used to derive the criteria determining the existence of T-periodic solutions for the heat equation being at resonance at infinity.

Dark soliton and periodic wave solutions of nonlinear evolution equations

Abstract This paper studies the Kudryashov-Sinelshchikov and Jimbo-Miwa equations. Subsequently, we formally derive the dark (topological) soliton solutions for these equations. By using the sine-cosine method, some additional periodic solutions are derived. The physical parameters in the soliton solutions of the ansatz method, amplitude, inverse width and velocity, are obtained as functions of the dependent model coefficients. PACS Codes:02.30.Jr, 02.70.Wz, 05.45.Yv, 94.05.Fg.