Elliptic Function Rational Expansion Method Research Articles

New exact travelling solutions of the generalized Hirota equation

In this paper, we solve the generalized Hirota equation with an external potential by means of the new Jacobi elliptic function rational expansion method and the exponential rational function method. The given methods are very effective to solve some nonlinear Schrödinger equations. Furthermore, some solutions are degraded to trigonometric solutions and solitary solutions in extreme cases.

Exact solutions and bifurcation for the resonant nonlinear Schrödinger equation with competing weakly nonlocal nonlinearity and fractional temporal evolution

ABSTRACT The resonant nonlinear Schrödinger equation (RNLSE) with competing weakly nonlocal nonlinearity and fractional temporal evolution, which describes the propagation of optical solitons along the nonlinear optical fibers, is investigated in this paper. Dynamic behavior of such equation with the parabolic-type nonlinearity is discussed. The relationship between the orbits of the dynamic system and traveling wave solutions is demonstrated. Particularly, a family of hyperbolic curves near the saddle points implies the existence of one family of the braking wave solutions correspondingly. Furthermore, the -expansion method and Jacobi elliptic function rational expansion method are conducted to get the exact traveling wave solutions, such as periodic wave solutions, shock wave solutions, and breaking wave solutions.

Solitons and other solutions of the nonlinear fractional Zoomeron equation

In order to investigate the nonlinear fractional Zoomeron equation, we propose three methods, namely the Jacobi elliptic function rational expansion method, the exponential rational function method and the new Jacobi elliptic function expansion method. Many kinds of solutions are obtained and the existence of these solutions is determined. For some parameters, these solutions can degenerate to the envelope shock wave solutions and the envelope solitary wave solutions. A comparison of our new results with the well-known results is made. The methods used here can also be applicable to other nonlinear partial differential equations. The fractional derivatives in this work are described in the modified Riemann–Liouville sense.

A note on “Exact solutions of the unstable nonlinear Schrödinger equation with the new Jacobi elliptic function rational expansion method and the exponential rational function method”

This manuscript presents a short note on the “Exact solutions of the unstable nonlinear Schrödinger equation with the new Jacobi elliptic function rational expansion method and the exponential rational function method”. The objective here is to give clarifications concerning some results in [1].

Exact solutions of the unstable nonlinear Schrödinger equation with the new Jacobi elliptic function rational expansion method and the exponential rational function method

In this paper, by using the new Jacobi elliptic function rational expansion method and the exponential rational function method, new exact solutions of the unstable nonlinear Schrödinger equation are obtained. The present methods are very effective and simples and can be applied to other types of nonlinear equations. The results found can be compared with other methods used to solve the same model.

New generalized Jacobi elliptic function rational expansion method

In this work, a new generalized Jacobi elliptic function rational expansion method is based upon twenty-four Jacobi elliptic functions and eight double periodic Weierstrass elliptic functions, which solve the elliptic equation ϕ ′ 2 = r + p ϕ 2 + q ϕ 4 , is described. As a consequence abundant new Jacobi–Weierstrass double periodic elliptic functions solutions for (3+1)-dimensional Kadmtsev–Petviashvili (KP) equation are obtained by using this method. We show that the new method can be also used to solve other nonlinear partial differential equations (NPDEs) in mathematical physics.

New Extended Jacobi Elliptic Function RationalExpansion Method and Its Application

In this paper, an extendedJacobi elliptic function rational expansion method is proposed forconstructing new forms of exact Jacobi elliptic function solutionsto nonlinear partial differential equations by means of making amore general transformation. For illustration, we apply the methodto the (2+1)-dimensional dispersive long wave equation andsuccessfully obtain many new doubly periodic solutions, whichdegenerate as soliton solutions when the modulus m approximates1. The method can also be applied to other nonlinear partialdifferential equations.

Further Extended Jacobi Elliptic Function Rational Expansion Method andNew Families of Jacobi Elliptic Function Solutions to (2+1)-Dimensional DispersiveLong Wave Equation

In this paper, a further extended Jacobi elliptic function rational expansionmethod is proposed for constructing new forms of exact solutionsto nonlinear partial differential equations by making a moregeneral transformation. For illustration, we apply the method to(2+1)-dimensional dispersive long wave equation andsuccessfully obtain many new doubly periodic solutions. When themodulus m→1, these solutions degenerate as solitonsolutions. The method can be also applied to other nonlinearpartial differential equations.

The extension of the Jacobi elliptic function rational expansion method

Using a new ansätz, we extend the Jacobi elliptic function rational expansion method and apply it to the asymmetric Nizhnik–Novikov–Veselov equations and the Davey–Stewartson equations. With the aid of symbolic computation, we construct more new Jacobi elliptic doubly periodic wave solutions and the corresponding solitary wave solutions and triangular functional (singly periodic) solutions.

The extended Jacobi elliptic function method to solve a generalized Hirota–Satsuma coupled KdV equations

Abstract In this paper, we extend the Jacobi elliptic function rational expansion method by using a new generalized ansatz. With the help of symbolic computation, we construct more new explicit exact solutions of nonlinear evolution equations (NLEEs). We apply this method to a generalized Hirota–Satsuma coupled KdV equations and gain more general solutions. The general solutions not only contain the solutions by the existing Jacobi elliptic function expansion methods but also contain many new solutions. When the modulus of the Jacobi elliptic functions m → 1 or 0, the corresponding solitary wave solutions and triangular functional (singly periodic) solutions are also obtained.

An extended Jacobi elliptic function rational expansion method and its application to ([formula omitted])-dimensional dispersive long wave equation

With the aid of computerized symbolic computation, a new elliptic function rational expansion method is presented by means of a new general ansatz, in which periodic solutions of nonlinear partial differential equations that can be expressed as a finite Laurent series of some of 12 Jacobi elliptic functions, is more powerful than exiting Jacobi elliptic function methods and is very powerful to uniformly construct more new exact periodic solutions in terms of rational formal Jacobi elliptic function solution of nonlinear partial differential equations. As an application of the method, we choose a (2+1)-dimensional dispersive long wave equation to illustrate the method. As a result, we can successfully obtain the solutions found by most existing Jacobi elliptic function methods and find other new and more general solutions at the same time. Of course, more shock wave solutions or solitary wave solutions can be gotten at their limit condition.

Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to (1 + 1)-dimensional dispersive long wave equation

Abstract Our Jacobi elliptic function rational expansion method is extended to be a more powerful method, called the extended Jacobi elliptic function rational expansion method, by using more general ansatz. The (1 + 1)-dimensional dispersive long wave equation is chosen to illustrate the approach. As a consequence, we can successfully obtain the solutions found by most existing Jacobi elliptic function methods and find other new and more general solutions at the same time. When the modulus m → 1, these doubly periodic solutions degenerate as soliton solutions. The method can be also applied to other nonlinear differential equations.

A new general algebraic method with symbolic computation to construct new doubly-periodic solutions of the (2 + 1)-dimensional dispersive long wave equation

For constructing more new exact doubly-periodic solutions in terms of rational form Jacobi elliptic function of nonlinear evolution equations, a new direct and unified algebraic method, named Jacobi elliptic function rational expansion method, is presented and implemented in a computer algebraic system. Compared with most of the existing Jacobi elliptic function expansion methods, the proposed method can be expected to obtain new and more general formal solutions. We choose a (2 + 1)-dimensional dispersive long wave equation to illustrate the method.

Jacobi Elliptic Function Rational Expansion Method with Symbolic Computation to Construct New Doubly-periodic Solutions of Nonlinear Evolution Equations

A new Jacobi elliptic function rational expansion method is presented by means of a new general ansatz and is very powerful, with aid of symbolic computation, to uniformly construct more new exact doubly-periodic solutions in terms of rational form Jacobi elliptic function of nonlinear evolution equations (NLEEs). We choose a (2+1)-dimensional dispersive long wave equation to illustrate the method. As a result, we obtain the solutions found by most existing Jacobi elliptic function expansion methods and find other new and more general solutions at the same time. When the modulus of the Jacobi elliptic functions m→1 or 0, the corresponding solitary wave solutions and trigonometric function (singly periodic) solutions are also found.

A new Jacobi elliptic function rational expansion method and its application to (1+1)-dimensional dispersive long wave equation

Abstract With the aid of computerized symbolic computation, a new elliptic function rational expansion method is presented by means of a new general ansatz and is very powerful to uniformly construct more new exact doubly-periodic solutions in terms of rational formal Jacobi elliptic function of nonlinear evolution equations (NLEEs). As an application of the method, we choose a (1 + 1)-dimensional dispersive long wave equation to illustrate the method. As a result, we can successfully obtain the solutions found by most existing Jacobi elliptic function methods and find other new and more general solutions at the same time. Of course, more shock wave solutions or solitary wave solutions can be gotten at their limit condition.