Bäcklund Transformations and Exact Solutions for Alfvén Solitons in a Relativistic Electron–Positron Plasma

Khater, A. H.; El-Kalaawy, Omar H.; Callebaut, D. K.

doi:10.1088/0031-8949/58/6/001

Cited by 94 publications

(53 citation statements)

References 31 publications

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“…where , , and are constants, which is a transformed generalization of the combined KdV-mKdV equation [33][34][35]. In order to solve (17) by the fractional mapping method, we use the traveling wave transformation ( , ) = ( ), = + , then, (17) is reduced to the following nonlinear FODE:…”

Section: The Space-time Fractional Combined Kdv-mkdv Equationmentioning

confidence: 99%

Analytic Solutions of the Space-Time Fractional Combined KdV-mKdV Equation

Abdel-Salam

Al-Muhiameed

2015

Mathematical Problems in Engineering

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The fractional mapping method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional combined KdV-mKdV equation. Many types of exact analytical solutions are obtained. The solutions include generalized trigonometric and hyperbolic functions solutions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time.

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Section: The Space-time Fractional Combined Kdv-mkdv Equationmentioning

confidence: 99%

Analytic Solutions of the Space-Time Fractional Combined KdV-mKdV Equation

Abdel-Salam

Al-Muhiameed

2015

Mathematical Problems in Engineering

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“…This equation is one of the two important equations in physical contexts (see [13,14,16,28,42]). For the positive MKdV equation, say, α > 0, β < 0, the N-soliton and N-pole solutions obtained by the Hirota method in [12,35] are related to the single and double real spectral parameter, respectively.…”

Section: Mkdv Equationmentioning

confidence: 99%

A new periodic solution to Jacobi elliptic functions of MKdV equation and BBM equation

Zhang

Deng

2012

Acta Math. Appl. Sin. Engl. Ser.

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Based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equations. New exact solutions to the Jacobi elliptic function of MKdV equations and Benjamin-Bona-Mahoney (BBM) equations are obtained with the aid of computer algebraic system Maple. The method is also valid for other (1+1)-dimensional and higher dimensional systems.Nonlinear partial differential equations (NPDEs) are widely used to describe phenomena in physics, biology, chemistry, and so on. Some of the most interesting features of physical systems are hidden in their nonlinear behavior, and can only be studied with appropriate methods. Over the past decades, many authors have mainly paid attention to construct various methods such as Bäcklund transformation (see [1,5,10,11,36]), Darboux transformation [36] , Inverse scattering method [10] , Hirota's bilinear method [11] , the tanh-function method [24] , the sine-cosine method [38], the homogeneous balance method [37] , symmetry reduction method (see [2, 19, 21-23, 26, 27, 29]), the geometrical method [6] , the formal variable separation approach [4] , the multilinear variable separation approach (MLVSA) [17,20] , the functional variable separation approach (FVSA) (see [30,31,39]) and the derivative-dependent functional variable separation approach (DDFVSA) [40] . Recently, an extended tanh-function method [7] and symbolic computation are used in [8] for solving the new coupled modified KdV equations to obtain four kinds of soliton solutions.The aim of this paper is to use a direct method to solve two different types of equations such as the MKdV and BBM equations.

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“…(14). Let us start with the general description of a threedimensional Riemannian manifold S following reference [28].…”

Section: On Equations Describing Pssmentioning

confidence: 99%

“…Khater et al [7,8] used the notion of a differential equation (DE) for a function u(x, t) that describes a pseudo-spherical surface (pss), and they derived some Bäcklund transformations and conservation laws for NLEEs which are integrability condition of sl (2, R)-valued linear problems [9][10][11][12][13][14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%