Links between some (2 + 1)-dimensional nonlinear evolution equations and Painlevé-II equations

Using the idea of transformation, some links between (2 + 1)-dimensional nonlinear evolution equations and the ordinary differential equations Painlevé-II equations has been illustrated. The Kadomtsev–Petviashvili (KP) equation, generalized (2 + 1)-dimensional break soliton equation and (2 + 1)-dimensional Boussinesq equation are researched. As a result, some new interesting results about these (2 + 1)-dimensional PDEs have been obtained, such as the exact solutions with arbitrary functions, rich rational solutions and the nontrivial Bäcklund transformations have been derived.     

New families of exact soliton-like solutions and dromion solutions to the (2+1)-dimensional higher-order Broer–Kaup equations

Using homogeneous balance method we obtain Bäcklund transformation (BT) and a linear partial differential equation of higher-order Broer–Kaup equations. As a result, new soliton-like solutions and new dromion solution and other exact solutions of (2 + 1)-dimensional higher-order Broer–Kaup equations are given. By analyzing a soliton-like solution, we get some dromions solutions. This method, which can be generalized to some (2 + 1)-dimensional nonlinear evolution equations, is simple and powerful.     

Comments on “The generalizing Riccati equation mapping method in nonlinear evolution equation: Application to (2 + 1)-dimensional Boiti–Leon–Pempinelle equation”

By means of a so-called generalizing Riccati equation mapping method, Zhu [Zhu S D, Chaos, Solitons & Fractals; 2006. doi:10.1016/j.chaos.2006.10.015] has claimed that abundant new solutions to the (2 + 1)-dimensional Boiti–Leon–Pempinelle (BLP) equation are derived. Based on the derived variable separation solution and by selecting appropriate functions, he has asserted that abundant new non-travelling waves are obtained. We show that the generalizing Riccati equation mapping method is equivalent to the usual mapping approach, and say nothing of the conclusion that many new non-travelling wave solutions have been found.     

A new generalized algebra method and its application in the (2 + 1) dimensional Boiti–Leon–Pempinelli equation

In the present paper, some types of general solutions of a first-order nonlinear ordinary differential equation with six degree are given and a new generalized algebra method is presented to find more exact solutions of nonlinear differential equations. As an application of the method and the solutions of this equation, we choose the (2 + 1) dimensional Boiti Leon Pempinelli equation to illustrate the validity and advantages of the method. As a consequence, more new types and general solutions are found which include rational solutions and irrational solutions and so on. The new method can also be applied to other nonlinear differential equations in mathematical physics.     

New generalized Jacobi elliptic function expansion method

A new generalized Jacobi elliptic function expansion method is described and used for constructing many new exact travelling wave solutions for nonlinear partial differential equations (PDEs) in a unified way. We obtain many new Jacobi and Weierstrass double periodic elliptic function solutions for (3 + 1)-dimensional Kadmtsev–Petviashvili (KP) equation. This method can be applied to many other equations.     

Pfaffian solutions and extended Pfaffian solutions to (3 + 1)-dimensional Jimbo–Miwa equation

Based on the Pfaffian derivative formula and Hirota bilinear method, the Pfaffian solutions to (3 + 1)-dimensional Jimbo–Miwa equation are obtained under a set of linear partial differential condition. Moreover, we extend the linear partial differential condition and proved that (3 + 1)-dimensional Jimbo–Miwa equation has extended Pfaffian solutions. As examples, special exact two-soliton solution and three-soliton solution are computed and plotted. Our results show that (3 + 1)-dimensional Jimbo–Miwa equation has Pfaffian solutions like BKP equation.     

New exact solutions to MKDV-Burgers equation and (2 + 1)-dimensional dispersive long wave equation via extended Riccati equation method

In this paper, with the aid of symbolic computation and a general ansätz, we presented a new extended rational expansion method to construct new rational formal exact solutions to nonlinear partial differential equations. In order to illustrate the effectiveness of this method, we apply it to the MKDV-Burgers equation and the (2 + 1)-dimensional dispersive long wave equation, then several new kinds of exact solutions are successfully obtained by using the new ansätz. The method can also be applied to other nonlinear partial differential equations.     

Explicit exact solutions for (2 + 1)-dimensional Boiti–Leon–Pempinelli equation

In this paper, we construct explicit exact solutions for the coupled Boiti–Leon–Pempinelli equation (BLP equation) by using a extended tanh method and symbolic computation system Mathematica. By means of the method, many new exact travelling wave solutions for the BLP system are successfully obtained. the extended tanh method can be applied to other higher-dimensional coupled nonlinear evolution equations in mathematical physics.     

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.     

Variable-coefficient projective Riccati equation method and its application to a new (2 + 1)-dimensional simplified generalized Broer–Kaup system

In this paper, based on a new intermediate transformation, a variable-coefficient projective Riccati equation method is proposed. Being concise and straightforward, it is applied to a new (2 + 1)-dimensional simplified generalized Broer–Kaup (SGBK) system. As a result, several new families of exact soliton-like solutions are obtained, beyond the travelling wave. When imposing some condition on them, the new exact solitary wave solutions of the (2 + 1)-dimensional SGBK system are given. The method can be applied to other nonlinear evolution equations in mathematical physics.     

Type-II hidden symmetry and nonlinear self-adjointness of Boiti–Leon–Pempinelli equation

This paper focuses on two aspects. Firstly, we convert Boiti–Leon–Pempinelli (BLP) equation to (1+1)-dimensional partial differential equation via similarity transformation, and then analyze hidden symmetry of BLP equations via studying classical and nonclassical symmetries of the (1+1)-dimensional equations. As a byproduct, some new invariant solutions of BLP equations are constructed. Secondly, we show that BLP equation is nonlinearly self-adjoint and give the general formula of conservation laws.     

A generalized F-expansion method to find abundant families of Jacobi Elliptic Function solutions of the (2 + 1)-dimensional Nizhnik–Novikov–Veselov equation

In the present paper, a generalized F-expansion method is proposed by further studying the famous extended F-expansion method and using a generalized transformation to seek more types of solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose (2 + 1)-dimensional Nizhnik–Novikov–Veselov equations to illustrate the validity and advantages of the method. As a result, abundant new exact solutions are obtained including Jacobi Elliptic Function solutions, soliton-like solutions, trigonometric function solution etc. The method can be also applied to other nonlinear partial differential equations.     

On shifted periodic solutions of two nonlinear equations

Using the linear superposition approach, we find periodic solutions with shifted periods and velocities of the (2 + 1)-dimensional modified Zakharov–Kuznetsov equation and the (3 + 1)-dimensional Kadomtsev–Petviashvili equation by making appropriate linear superpositions of known periodic solutions. This unusual procedure of generating solutions of nonlinear evolution equations is successful as a consequence of some cyclic identities satisfied by the Jacobi elliptic functions which reduce by 2 (or a larger even number) the degree of cyclic homogeneous polynomials in Jacobi elliptic functions.     

New exact solutions to the Zakharov–Kuznetsov equation and its generalized form

In this paper, the extended hyperbolic function method is used for analytic treatment of the (2 + 1)-dimensional Zakharov–Kuznetsov (ZK) equation and its generalized form. We can obtained some new explicit exact solitary wave solutions, the multiple nontrivial exact periodic travelling wave solutions, the solitons solutions and complex solutions. Some known results in the literatures can be regarded as special cases. The methods employed here can also be used to solve a large class of nonlinear evolution equations.     

New exact traveling wave solutions of the (3 + 1) dimensional Kadomtsev–Petviashvili (KP) equation

The repeated homogeneous balance method is used to construct new exact traveling wave solutions of the (3 + 1) dimensional Kadomtsev–Petviashvili (KP) equation, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation, respectively. Many new exact traveling wave solutions are successfully obtained, which contain soliton-like and periodic-like solutions. This method is straightforward and concise, and it can be also applied to other nonlinear evolution equations.     

Symbolic computation and new families of exact non-travelling wave solutions of (2 + 1)-dimensional Konopelchenko–Dubrovsky equations

The improved tanh function method [Chaos, Solitons & Fractals 2005;24:257] is further improved by constructing new ansatz solution of the considered equation. As its application, the (2 + 1)-dimensional Konopelchenko–Dubrovsky equations are considered and abundant new exact non-travelling wave solutions are obtained.     

Symmetry reduction and new non-traveling wave solutions of (2 + 1)-dimensional breaking soliton equation

In this paper, the new idea of a combination of Lie group method and homoclinic test technique is first proposed to seek non-traveling wave solutions of (2 + 1)-dimensional breaking soliton equation. The system is reduced to some (1 + 1)-dimensional nonlinear equations by applying the Lie group method and solves reduced equation with homoclinic test technique. Based on this idea and with the aid of symbolic computation, some new explicit solutions of similar systems can be obtained.     

An improved generalized F-expansion method and its application to the (2 + 1)-dimensional KdV equations

An improved generalized F-expansion method is proposed to seek exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the (2 + 1)-dimensional KdV equations to illustrate the validity and advantages of the proposed method. Many new and more general non-travelling wave solutions are obtained, including single and combined non-degenerate Jacobi elliptic function solutions, soliton-like solutions, trigonometric function solutions, each of which contains two arbitrary functions.     

New abundant exact solutions for the (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov system

In this paper, the extended hyperbolic function method is used for analytic treatment of the (2 + 1)-dimensional generalized Nizhnik–Novikov–Veselov (GNNV) system. We can obtained some new explicit exact solitary wave solutions, the multiple nontrivial exact periodic travelling wave solutions, the soliton solutions and complex solutions. Some known results in the literatures can be regarded as special cases. The methods employed here can also be used to solve a large class of nonlinear evolution equations.     

A series of new exact solutions of nonlinear evolution equations

With the aid of computer symbolic computation system Maple, the generalized auxiliary equation method is first applied to two nonlinear evolution equations, namely, the nonlinear elastic rod equation and (2 + 1)‐dimensional Boiti‐Leon‐Pempinelli equation. As a results, some new types of exact traveling wave solutions are obtained which include bell and kink profile solitary wave solutions, and triangular periodic wave solutions and singular solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010