Jacobi Elliptic Function Expansion Method Research Articles

Forced dissipative Boussinesq equation for solitary waves excited by unstable topography

In the paper, the effects of topographic forcing and dissipation on solitary Rossby waves are studied. Special attention is given to solitary Rossby waves excited by unstable topography. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a forced dissipative Boussinesq equation. By using the modified Jacobi elliptic function expansion method and the pseudo-spectral method, the solutions of homogeneous and inhomogeneous dissipative Boussinesq equation are obtained, respectively. With the help of these solutions, the evolutional character of Rossby waves under the influence of dissipation and unstable topography is discussed.

Jacobi doubly periodic wave solutions for three versions of Benjamin-Bona- Mahony equation

A variety of exact solutions for the Kadomtsov-Petviashvili- Benjamin-Bona-Mahony (KP-BBM) equation, nonlinear Zakharov-Kuznetsov- Benjamin-Bona-Mahony (ZK-BBM) equation and the generalized ZK-BBM equation are developed by means of the extended Jacobi elliptic function expansion method. Soliton and triangular periodic solutions can be established as the limits of Jacobi doubly periodic wave solutions. Key words: Jacobi elliptic function method, soliton and triangular periodic solutions, nonlinear dispersive Kadomtsov-Petviashvili- Benjamin-Bona-Mahony (KP-BBM) equation, nonlinear Zakharov-Kuznetsov- Benjamin-Bona-Mahony (ZK-BBM) equation, the generalized ZK-BBM equation.

New exact Jacobi elliptic solutions for the generalized time-dependent variable coefficients KdV- mKdV equation

In this article, we use the improved general mapping deformation method based on the generalized Jacobi elliptic functions expansion method with computerized symbolic computation to construct more new exact solutions of a generalized time-dependent variable coefficients KdV- mKdV equation. As a result, new generalized Jacobi elliptic function-like solutions, soliton-like solutions and trigonometric function solutions are obtained by using this method. This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics. Key words: Exact solutions, generalized time-dependent variable coefficients KdV- mKdV equation, improved general mapping deformation method, Jacobi elliptic functions.

A Note on Exact Traveling Wave Solutions of the Perturbed Nonlinear Schrödinger's Equation with Kerr Law Nonlinearity

In this paper, we investigate nonlinear the perturbed nonlinear Schrödinger's equation (NLSE) with Kerr law nonlinearity given in [Z.Y. Zhang, et al., Appl. Math. Comput. 216 (2010) 3064] and obtain exact traveling solutions by using infinite series method (ISM), Cosine-function method (CFM). We show that the solutions by using ISM and CFM are equal. Finally, we obtain abundant exact traveling wave solutions of NLSE by using Jacobi elliptic function expansion method (JEFEM).

Exact traveling wave solutions of modified zakharov equations for plasmas with a quantum correction

In this article, the authors study the exact traveling wave solutions of modified Zakharov equations for plasmas with a quantum correction by hyperbolic tangent function expansion method, hyperbolic secant expansion method, and Jacobi elliptic function expansion method. They obtain more exact traveling wave solutions including trigonometric function solutions, rational function solutions, and more generally solitary waves, which are called classical bright soliton, W-shaped soliton, and M-shaped soliton.

A New Extended Jacobi Elliptic Function Expansion Method and Its Application to the Generalized Shallow Water Wave Equation

With the aid of symbolic computation, a new extended Jacobi elliptic function expansion method is presented by means of a new ansatz, in which periodic solutions of nonlinear evolution equations, which can be expressed as a finite Laurent series of some 12 Jacobi elliptic functions, are very effective to uniformly construct more new exact periodic solutions in terms of Jacobi elliptic function solutions of nonlinear partial differential equations. As an application of the method, we choose the generalized shallow water wave (GSWW) equation to illustrate the method. As a result, we can successfully obtain more new solutions. Of course, more shock wave solutions or solitary wave solutions can be gotten at their limit condition.

New Jacobi elliptic function-like solutions for the general KdV equation with variable coefficients

In this paper, an improved general mapping deformation method based on the generalized Jacobi elliptic functions expansion method with computerized symbolic computation is used to construct more new exact solutions of a generalized KdV equation with variable coefficients. As a result, seven families of new generalized Jacobi elliptic function-like solutions, soliton-like solutions and trigonometric function solutions of the equation are obtained by using this method, which shows that the general method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.

Combined periodic wave and solitary wave solutions in two-component Bose-Einstein condensates

In this paper, the Jacobi elliptic function expansion method provides an effective approach to obtain the exact periodic wave solutions of two-component Bose-Einstein condensates. Exact combined bright-bright and dark-dark soliton wave solutions can be achieved in their limit conditions. We also obtain the different formation regions of combined solitons. Our results show that the intraspecies (interspecies) interaction strengths clearly affect the formation of dark-dark, bright-bright and dark-bright soliton solutions in different regions.

New special structures to the (2+1)-dimensional breaking soliton equations

Applying the Jacobi elliptic function expansion method to the (2+1)-dimensional breaking soliton equations (BSE for short), we derive some types of exact solutions to the BSE. Based on the derived solutions, we obtain some special structures, such as spatially localized or periodic excitations. For specific choices, we show some features of the (2+1)-dimensional BSE, such as breathers or breather lattice solutions.

Variable-coefficient Jacobi elliptic function expansion method for (2+1)-dimensional Nizhnik–Novikov–Vesselov equations

Abstract In this paper, a variable-coefficient Jacobi elliptic function expansion method is proposed to seek more general exact solutions of nonlinear partial differential equations. Being concise and straightforward, this method is applied to the (2+1)-dimensional Nizhnik–Novikov–Vesselov equations. As a result, many new and more general exact non-travelling wave and coefficient function solutions are obtained including Jacobi elliptic function solutions, soliton-like solutions and trigonometric function solutions. To give more physical insights to the obtained solutions, we present graphically their representative structures by setting the arbitrary functions in the solutions as specific functions.

Exact Solutions of Bose-Einstein Condensate in Linear Magnetic Field and Time-Dependent Laser Field

Nonlinear evolution equations (NLEEs) are frequently used to model a wide variety of phenomena in physics, chemistry, biology and even in social science field. Seeking and constructing exact solutions especially soliton solutions of NLEEs is one of the most important and essential task in nonlinear science. With the help of exact solutions, when they exist, the nonlinear phenomena can be better understood. In the past decades, various sophisticated methods have been created such as the inverse scattering method [1], the Backlund transformations [2], the Darboux transformations [3] and the Hirota bilinear method [4]. Especially, in recent years, with the rapid development of computerized symbolic computation, many direct and effective algebraic methods are proposed such as the homogeneous balance method [5], tanh function method [6], the Jacobi elliptic function expansion method [7], the F-expansion techniques [8], exp-function method [9] and so on. Recently, Wang et al. [10] introduced a new direct method called G ′ G -expansion method for a reliable treatment of the NLEEs. Then this method is generalized and widely used to seek travelling wave solutions, non-travelling wave solutions and coefficient function solutions for many nonlinear physical models [11–14], and even extended to solve NLEE with variable coefficients [15]. The Gross–Pitaevskii equation appears as a relevant model in various areas of physics: nonlinear optics, fluid mechanics, Bose–Einstein condensation (BEC). The realization of BEC of weakly interacting atomic gases stim-

Cas de légionellose en milieu professionnel

In this paper, author employed Jacobi elliptic function expansion method to build the new wave solutions of time fractional modified Camassa–Holm equation which is completely integrable dispersive shallow-water equation. In ocean engineering, Camassa–Holm equation is generally used as a tool in computer simulation of the water waves in shallow sees, coastal and harbors. The obtained solutions show that the Jacobi elliptic function expansion method (JEFEM) which based on Jacobi elliptic functions is an efficient, reliable, applicable and accurate tool for analytic approximation of a wide variety of nonlinear conformable time fractional partial differential equations.

Extended Jacobi Elliptic Function Expansion Method to the ZK‐MEW Equation

The extended Jacobi elliptic function expansion method is applied for Zakharov‐Kuznetsov‐modified equal‐width (ZK‐MEW) equation. With the aid of symbolic computation, we construct some new Jacobi elliptic doubly periodic wave solutions and the corresponding solitary wave solutions and triangular functional (singly periodic) solutions.

Jacobi Elliptic Function Solutions for (2 + 1) Dimensional Boussinesq and Kadomtsev-Petviashvili Equation

(2 + 1) dimensional Boussinesq and Kadomtsev-Petviashvili equation are investigated by employing Jacobi elliptic function expansion method in this paper. As a result, some new forms traveling wave solutions of the equation are reported. Numerical simulation results are shown. These new solutions may be important for the explanation of some practical physical problems. The results of this paper show that Jacobi elliptic function method can be a useful tool in obtaining evolution solutions of nonlinear system.

Extended Jacobi elliptic function expansion solutions of variant Boussinesq equations

With the aid of symbolic computation Maple, an extended Jacobi elliptic function expansion method is presented and successfully applied to variant Boussinesq equations. As a result, abundant periodic wave solutions in terms of the Jacobi elliptic functions are obtained. When the modulus m → 1 or m → 0, exact solitary wave solutions and trigonometric function solutions are also derived. The properties of four new solutions are graphically studied.

Optical solitons in presence of higher-order effects

In this paper, we seek special soliton solutions of so-called Higher-Order Nonlinear Schroedinger Equation, which describes the propagation of light waves in nonlinear optical media. Using a developed Jacobi elliptic function expansion method described in [8], we show that in the case of four-order dispersion these traveling wave solutions do not exist. As a byproduct, we regain the results obtained by other authors [8, 9] for the case when only the dispersion to the third order is taken into account. Full Text: PDF References C. Long Van, M. Trippenbach, "New Optical Solitary waves in the higher order nonlinear Schroedinger Equation", Scientific Bulletin 126, Technical University of Zielona Gora, Poland 2001, p.19 (Cao Long Van ed.). V. Cao Long, H. Nguyen Viet, M. Trippenbach, K. Dinh Xuan,"Propagation Technique for Ultrashort Pulses I", Computational Methods in Science and Technology 14(1), 5 (2008) V. Cao Long, H. Nguyen Viet, M. Trippenbach, K. Dinh Xuan,"Propagation Technique for Ultrashort Pulses II ", Computational Methods in Science and Technology 14(1), 13 (2008) V. Cao Long, H. Nguyen Viet, M. Trippenbach, K. Dinh Xuan,"Propagation Technique for Ultrashort Pulses III ", Computational Methods in Science and Technology 14(1), 21 (2008) C. Long Van, P. P. Goldstein, "A concise Course in nonlinear Partial Differential Equations" (University of Zielona Gora 2008). A. V. Buryak and Yu. S. Kivshar,"Solitons due to second harmonic generation", Phys. Lett. A 197, 407 (1995) [CrossRef] M. Trippenbach, M. Matuszeski, E. Infreld, Cao Long Van, R. S. Tasgal, J. B. Band,"Enhancement of third harmonic generation by wave vector mismatch to counter phase- modulation ", Optics Comm. 229, 391 (2004) [CrossRef] S. A. El-Wakil, M.A. Abdou, A. Elhanbaly,"New solitons and periodic wave solutions for nonlinear evolution equations", Phys. Lett. A 353, 40 (2006) [CrossRef] Q. Yang, C.Q. Dai, J. F. Zhang,"Higher-Order Effects Induced Optical Solitons in Fiber", International Journal of Theoretical Physics 44, 1117 (2005) [CrossRef]

New exact Jacobi elliptic functions solutions for the generalized coupled Hirota–Satsuma KdV system

In this paper, an generalized Jacobi elliptic functions expansion method with computerized symbolic computation is used for constructing more new exact Jacobi elliptic functions solutions of the generalized coupled Hirota–Satsuma KdV system. As a result, eight families of new doubly periodic solutions are obtained by using this method, some of these solutions are degenerated to solitary wave solutions and triangular functions solutions in the limit cases when the modulus of the Jacobi elliptic functions m → 1 or 0, which shows that the applied method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.

PERIODIC WAVE SOLUTIONS TO MODIFIED NONLINEAR SCHRÖDINGER EQUATION PERTAINING TO NEGATIVE INDEX MATERIALS

In this paper, we intend to determine periodic wave solutions for the modified nonlinear Schrödinger equation pertaining to negative index materials. We have treated the propagation equation possessing higher order linear and nonlinear dispersion terms with Jacobian elliptic function expansion method and arrived at the Jacobian elliptic periodic wave solutions. When the module of the Jacobian elliptic function m → 1, these solutions degenerate to the solitary wave solutions of the governing system.

New Jacobi elliptic functions solutions for the variable-coefficient mKdV equation

In this work, a new generalized Jacobi elliptic functions expansion method based upon four new Jacobi elliptic functions is described and abundant new Jacobi-like elliptic functions solutions for the variable-coefficient mKdV equation are obtained by using this method, some of these solutions are degenerated to solitary-like solutions and triangular-like functions solutions in the limit cases when the modulus of the Jacobi elliptic functions m → 1 or 0, which shows that the new method can be also used to solve other nonlinear partial differential equations in mathematical physics.

Discrete Jacobi elliptic function expansion method for nonlinear differential-difference equations

In this paper, an improved algorithm is devised to derive exact travelling wave solutions of nonlinear differential-difference equations (DDEs) by means of Jacobi elliptic functions. With the aid of symbolic computation, we choose the integrable discrete nonlinear Schrödinger equation to illustrate the validity and advantages of the method. As a result, new and more general Jacobi elliptic function solutions are obtained, from which hyperbolic function solutions and trigonometric function solutions are derived when the modulus m→1 and 0. It is shown that the proposed method provides a more effective mathematical tool for nonlinear DDEs in mathematical physics.