Periodic Form Solutions Research Articles

Extended rational expansion method for differential-difference equation

In this paper, the rational formal expansion method is extended to construct a series of soliton-like and period-form solutions for nonlinear differential-difference equations. The efficiency of the method has been demonstrated on Volterra Lattice. Compared with most existing methods, the proposed method not only recover some known solutions, but also find some new and more general solutions.

Application of Rational Expansion Method for Differential-Difference Equation

In this paper, we applied the rational formal expansion method to construct a series of soliton-like and period-form solutions for nonlinear differential-difference equations. Compared with most existing methods, the proposed method not only recovers some known solutions, but also finds some new and more general solutions. The efficiency of the method can be demonstrated on Toda Lattice and Ablowitz—Ladik Lattice.

New exact kink solutions and periodic form solutions for a generalized Zakharov–Kuznetsov equation with variable coefficients

In this short letter, with the aid of symbolic computational system Mathematic, new exact solutions including kink solutions, soliton-like solutions and periodic form solutions for a generalized Zakharov–Kuznetsov equation with variable coefficients are obtained using the generalized Riccati equation mapping method.

New exact kink solutions, solitons and periodic form solutions for a combined KdV and Schwarzian KdV equation

In this short letter, new exact solutions including kink solutions, soliton-like solutions and periodic form solutions for a combined version of the potential KdV equation and the Schwarzian KdV equation are obtained using the generalized Riccati equation mapping method.

Abundant new exact solutions for the [formula omitted]-dimensional Jimbo–Miwa equation

In this paper, by using the generalized Riccati equation mapping method, and picking up its new solutions authors obtain abundant new exact solutions including kink solutions, periodic form solutions, soliton-like solutions and rational solutions to a ( 3 + 1 ) -dimensional Jimbo–Miwa equation, respectively.

Some new solutions derived from the nonlinear (2+1)-dimensional Toda equation–an efficient method of creating solutions

This paper presents a new and efficient approach for constructing exact solutions to nonlinear differential–difference equations (NLDDEs) and lattice equation. By using this method via symbolic computation system MAPLE, we obtained abundant soliton-like and/or period-form solutions to the (2+1)-dimensional Toda equation. It seems that solitary wave solutions are merely special cases in one family. Furthermore, the method can also be applied to other nonlinear differential–difference equations.

Soliton-like and periodic form solutions to (2+1)-dimensional Toda equation

In this paper, we present a further extended tanh method for constructing exact solutions to nonlinear difference-differential equation(s) (NDDEs) and Lattice equations. By using this method via symbolic computation system MAPLE , we obtain abundant soliton-like and period-form solutions to the (2 + 1)-dimensional Toda equation. Solitary wave solutions are merely a special case in one family. This method can also be used to other nonlinear difference differential equations.

New Method for Finding a Series of Exact Solutions to Generalized Breaking Soliton Equation**The project supported by National Natural Science Foundation ofChina and the Natural Science Foundation of Shandong Province of China

In this paper, a new generalized extended tanh-functionmethod is presented for constructing soliton-like, period-form solutions ofnonlinear evolution equations (NEEs). This method is more powerful than theextended tanh-function method [Phys. Lett. A 277 (2000) 212] and the modifiedextended tanh-function method [Phys. Lett. A 285 (2001) 355]. Abundant newfamilies of the exact solutions of Bogoyavlenskii's generalized breakingsoliton equation are obtained by using this method and symbolic computationsystem Maple.

Generalized extended tanh-function method and its application

In this paper, a new generalized extended tanh-function method is presented for constructing soliton-like, period-form solutions of nonlinear evolution equations (NEEs). Compared with most of the existing tanh-function method, extended tanh-function method, the modified extended tanh-function method and generalized hyperbolic-function method, the proposed method is more powerful. By using this method, we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some NEEs. Make use of the method, we study the (3 + 1)-dimensional potential-YTSF equation and obtain rich new families of the exact solutions, including the non-travelling wave and coefficient functions’ soliton-like solutions, singular soliton-like solutions, periodic form solutions.

New Families of Exact Solutions for (2+1)-DimensionalBroer–Kaup System

Using a further modified extended tanh-function method,rich new families of the exact solutions for the (2+1)-dimensionalBroer–Kaup (BK) system, comprising the non-traveling wave and coefficientfunctions' soliton-like solutions, singular soliton-like solutions, periodicform solutions, are obtained.

New Exact Solutions to (2+1)-Dimensional Dispersive Long Wave Equations**The project supported by the State Key Basic Research Program of China under Grant No. 1998030600 and National Natural Science Foundation of China under Grant Nos. 10072013 and 10072189

Based upon a further extended tanh method [Phys. Lett. A307 (2003) 269; Chaos, Solitons and Fractals 17 (2003) 669] and the symbolic computation system, Maple, we consider the (2+1)-dimensional dispersive long wave equations. We obtain many new solutions of the equation. These solutions contain soliton-like solutions, periodic form solutions, and some rational solutions.

Soliton-like solutions and periodic form solutions for two variable-coefficient evolution equations using symbolic computation

Some variable-coefficient generalizations of some nonlinear evolution equations (NLEEs) bear more realistic physical importance. By means of a generalized Riccati equation expansion (GREE) method and a symbolic computation system – Maple – we investigate the variable-coefficient Fisher-type equation and the nearly concentric KdV equation. As a result, rich families of exact analytic solutions for these two equations, including the non-travelling wave’s and coefficient functions’ soliton-like solutions, singular soliton-like solutions, and periodic form solutions, are obtained.

A New Modified Extended Tanh-Function Method and Its Application

In this paper, a new modified extended tanh-function method is presented for constructing soliton-like, periodic form solutions of nonlinear evolution equation (NEEs). This method is more powerful than the extended tanh-function method [Phys. Lett. A277 (2000) 212] and the modified extended tanh-function method [Phys. Lett. A285 (2001) 355]. Abundant new solutions of two physically important NEEs are obtained by using this method and symbolic computation system Maple.

Explicit Exact Solutions for the (2+1)-Dimensional Higher-Order Broer–Kaup System**The project supported by National Natural Science Foundation of China and the Natural Science Foundation of Shandong Province of China

Using the modified extended tanh-function method, explicit and exact traveling wave solutions for the (2+1)-dimensional higher-order Broer–Kaup (HBK) system, comprising new soliton-like and period-form solutions, are obtained.

Applications of a further extended tanh method

By using a further extended tanh method [Chaos Solitons Fract. 17 (2003) 669] and symbolic computation system Maple, we obtain abundant new explicit exact solutions of the (2 + 1)-dimensional Burgers equation and the (2 + 1)-dimensional Boussinesq equation. The new solutions including soliton-like solutions, period form solutions and some other solutions.

Generalized Riccati equation expansion method and its application to the Bogoyavlenskii's generalized breaking soliton equation

Based on the computerized symbolic system Maple and a Riccati equation, a Riccati equation expansion method is presented by a general ansatz. Compared with most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method and generalized hyperbolic-function method, the proposed method is more powerful. By use of the method, we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some nonlinear differential equations. Making use of the method, we study the Bogoyavlenskii's generalized breaking soliton equation and obtain rich new families of the exact solutions, including the non-travelling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions.

Soliton-like solutions for a (2+1) -dimensional nonintegrable KdV equation and a variable-coefficient KdV equation

Based on a Riccati equation and a symbolic computation system-Maple, a generalized Riccati equation expansion method is presented for constructing soliton-like solutions and periodic form solutions for some nonlinear evolution equations (NEEs) or NEEs with variable coefficients. Compared with most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method and the generalized hyperbolic-function method, the proposed method is more powerful. We study a (2+1)-dimensional general nonintegrable KdV equation, a KdV equation with variable coefficients. As a result, rich new families of exact solutions, including the non-travelling wave's and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions, are obtained. When setting the arbitrary functions in some solutions be equal to special constants or special functions, the solitary wave solutions can be recovered.

Generalized Riccati equation expansion method and its application to the (3 + 1)-dimensional Jumbo–Miwa equation

Based on the computerized symbolic system Maple and a Riccati equation, a generalized Riccati equation expansion method for constructing soliton-like solutions of non-linear evolution equations (NEEs) is presented by a new general ansätz. The proposed method is more powerful than most of the existing tanh methods, the extended tanh-function method, the modified extended tanh-function method and generalized hyperbolic-function method. By use of the method, we not only can successfully recover the previously known formal solutions but also construct new and more general formal solutions for some NEEs. Making use of the method, we study the the (3 + 1)-dimensional Jumbo–Miwa equation and obtain rich new families of the exact solutions, including the non-travelling wave and coefficient functions' soliton-like solutions, singular soliton-like solutions, periodic form solutions.

On a New Modified Extended Tanh-Function Method**The project supported by the National Key Basic Research Development of China (Grant No. 1998030600) and the National Natural Science Foundation of China (Grant No. 10072013)

In this paper, a new modified extended tanh-function method is presented for constructing multiple soliton-like, periodic form and rational solutions of nonlinear evolution equations (NLEEs). This method is more powerful than the extended tanh-function method [Phys. Lett. A 277 (2000) 212] and the modified extended tanh-function method [Phys. Lett. A 299 (2002) 179]. Abundant new solutions of two physically important NLEEs are obtained by using this method and symbolic computation system Maple.

Symbolic Computation and Construction of Soliton-like Solutions of some Nonlinear Evolution Equations

Based on the computerized symbolic computation system Maple and a Riccati equation, a new generalized Riccati equation expansion method for constructing non-travelling waves and coefficient functions’ soliton-like solutions of nonlinear evolution equations (NEEs) is presented by a general ansatz. Compared with most of the existing tanh methods, namely the extended tanh-function method, the modified extended tanh-function method and the generalized hyperbolic-function method, the proposed method is more powerful. By use of the method one can not only successfully recover the previously known formal solutions but also construct new and more general formal solutions for some NEEs. The cylindrical Korteweg-de Vries (CkdV) equation, a Potential Kadomstev-Petviashvili (PKP) equation, the two-dimensional KdV-Burgers equation are chosen to illustrate our method such that rich new families of exact solutions, including the non-travelling wave soliton-like solutions, singular soliton-like solutions, periodic form solutions are obtained. When taking arbitrary functions of the solutions as some special constants, the known travelling wave solutions of the PKP equation, two-dimensional KdV-Burgers equation can be recovered.