New exact solutions of the Broer–Kaup equations in (2 + 1)-dimensional spaces

With the aid of Maple, several new kinds of exact solutions for the Broer–Kaup equations in (2 + 1)-dimensional spaces are obtained by using a new ansätz. This approach can also be applied to other nonlinear evolution equations.     

Explicit exact solutions for the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

In this paper, we construct new explicit exact solutions for the coupled the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation (KD equation) by using a improved mapping approach and variable separation method. By means of the method, new types of variable-separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) for the KD system are successfully obtained. The improved mapping approach and variable separation method can be applied to other higher-dimensional coupled nonlinear evolution equations.     

A modified variable-coefficient projective Riccati equation method and its application to (2 + 1)-dimensional simplified generalized Broer–Kaup system

A modified variable-coefficient projective Riccati equation method is proposed and applied to a (2 + 1)-dimensional simplified and generalized Broer–Kaup system. It is shown that the method presented by Huang and Zhang [Huang DJ, Zhang HQ. Chaos, Solitons & Fractals 2005; 23:601] is a special case of our method. The results obtained in the paper include many new formal solutions besides the all solutions found by Huang and Zhang.     

Doubly periodic wave and folded solitary wave solutions for (2 + 1)-dimensional higher-order Broer–Kaup equation

A general solution including three arbitrary functions is obtained for the (2 + 1)-dimensional high-order Broer–Kaup equation by means of WTC truncation method. From the general solution, doubly periodic wave solutions in terms of the Jacobian elliptic functions with different modulus and folded solitary wave solutions determined by appropriate multiple valued functions are obtained. Some interesting novel features and interaction properties of these exact solutions and coherent localized structures are revealed.     

Solving the (2 + 1)-dimensional higher order Broer–Kaup system via a transformation and tanh-function method

In this paper, the (2 + 1)-dimensional higher order Broer–Kaup system is reduced to a simple nonlinear partial differential equation by a transformation, and utilizing tanh-function method we obtain many new exact solutions for the (2 + 1)-dimensional higher order Broer–Kaup system.     

Compacton,peakon and folded localized excitations for the (2 + 1)-dimensional Broer–Kaup system

In high dimensions there are abundant coherent soliton excitations. From the variable separation solutions for the (2 + 1)-dimensional Broer–Kaup system, three kinds of new localized excitations in this system are obtained. Some interesting novel features of these structures are revealed.     

The periodic wave solutions for the (2 + 1)-dimensional Konopelchenko–Dubrovsky equations

More periodic wave solutions expressed by Jacobi elliptic functions for the (2 + 1)-dimensional Konopelchenko–Dubrovsky equations are obtained by using the extended F-expansion method. In the limit cases, the solitary wave solutions and trigonometric function solutions for the equations are also obtained.     

Bäcklund transformation and multi-soliton solutions for a (2 + 1)-dimensional Korteweg-de Vries system via symbolic computation

In this paper, the (2 + 1)-dimensional Korteweg-de Vries system is symbolically investigated. By the bilinear method, the N-soliton solution is presented. Then, based on the Bäcklund transformation in bilinear form, a new Bäcklund transformation is obtained and new representation of the N-soliton solution is derived. A class of novel multi-soliton solutions are obtained by the new Bäcklund transformation and the availability of symbolic computation is demonstrated.     

Analytic solutions of a (2 + 1)‐dimensional variable‐coefficient Broer–Kaup system

Based on a Riccati equation and one of its new generalized solitary solutions constructed by the Exp‐function method, new analytic solutions with free parameters and arbitrary functions of a (2 + 1)‐dimensional variable‐coefficient Broer–Kaup system are obtained. These free parameters and arbitrary functions reveal that the (2 + 1)‐dimensional variable‐coefficient Broer–Kaup system has rich spatial structures. As an illustrative example, two new spatial structures are shown by setting the arbitrary functions as different Jacobi elliptic functions. Compared with tanh‐function method and its extensions, the method proposed in this paper is more powerful and it can be applied to other nonlinear evolution equations. Copyright © 2010 John Wiley & Sons, Ltd.     

Optimal solutions of a (3 + 1)-dimensional B-Kadomtsev-Petviashvii equation

We explored and specialized new Lie infinitesimals for the (3 + 1)-dimensional B-Kadomtsev-Petviashvii (BKP) using the commutation product, which results a system of nonlinear ODEs manually solved. Through two stages of Lie symmetry reduction, (3 + 1)-dimensional BKP equation is reduced to nonsolvable nonlinear ODEs using various combinations of optimal Lie vectors. Using the integration and Riccati equation methods, we investigate new analytical solutions for these ODEs. Back substituting to the original variables generates new solutions for BKP. Some selected solutions illustrated through three-dimensional plots.     

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.     

Multiple soliton solutions for a (2 + 1)-dimensional integrable KdV6 equation

In this work, a completely integrable (2 + 1)-dimensional KdV6 equation is investigated. The Cole-Hopf transformation method combined with the Hirota’s bilinear sense are used to determine two sets of solutions for this equation. Multiple soliton solutions are formally derived to emphasize its complete integrability. Moreover, multiple singular soliton solutions are also developed for this equation. The resonance relation for this equation does not exist.     

Chaotic behaviors in the (2 + 1)-dimensional breaking soliton system

Starting from the extended tanh-function method based on mapping method, the variable separation solutions of the (2 + 1)-dimensional breaking soliton system are derived. By further studying, we find that these variable separation solutions, which seem independent, actually depend on each other. Based on the derived variable separation solution, chaotic behaviors, i.e. periodic solution with chaotic behavior and chaotic peaked and compact line solitons, are investigated.     

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.     

Stochastic exact solutions to (2 + 1)-dimensional stochastic Borer–Kaup equation

(2 + 1)-dimensional Wick-type stochastic Borer–Kaup equations are researched by homogeneous balance method and tanh-function method. And some stochastic exact solutions of (2 + 1)-dimensional Wick-type stochastic Borer–Kaup equations are obtained via Hermite transformation.     

Symbolic computation of exact solutions for the (2 + 1)-dimensional generalized stochastic KP equation

In this paper, the F-expansion method is extended and applied to construct the exact solutions of the (2 + 1)-dimensional generalized Wick-type stochastic Kadomtsev–Petviashvili equation by the aid of the symbolic computation system Maple. Some new stochastic exact solutions which include kink-shaped soliton solution, singular soliton solution and triangular periodic solutions are obtained via this method and Hermite transformation.     

New exact solutions and fractal patterns of generalized Broer–Kaup system via a mapping approach

With the help of an extended mapping approach, a new type of variable separation solution with two arbitrary functions of the (2 + 1)-dimensional generalized Broer–Kaup (GBK) system is derived. Based on the derived solitary wave excitation, we reveal some regular fractal and stochastic fractal patterns in the (2 + 1)-dimensional GBK system.     

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.