Explicit solutions of nonlinear (2 + 1)-dimensional dispersive long wave equation

1 Introduction

Searching for explicit solutions of nonlinear evolution equations by using various methods has become the main goal for many authors, and many powerful methods to construct exact solutions of nonlinear evolution equations have been established and developed such as the tanh-function expansion and its various extension, the Jacobi elliptic function expansion (Inc and Evans, 2004; Liu et al., 2001; Yan, 2003; Yan and Zhang, 1999; Zayed et al., 2005, 2007; Abdou, 2007; Fan, 2000; Bekir, 2008; Chow, 1995; Zhang et al., 2008a,b), very recently, Wang et al. introduced a new method called the $\frac{G^{\prime }}{G}$ -expansion method (Wang et al., 2008) to look for travelling wave solutions of nonlinear evolution equations. The $\frac{G^{\prime }}{G}$ -expansion method is based on the assumptions that the travelling wave solutions can be expressed by a polynomial in $\frac{G^{\prime }}{G}$ , and that $G=G(\xi )$ satisfies a second order linear ordinary differential equation (ODE). The objective of this paper is to use a new method which is called the $(G^{\prime }/G)$ -expansion method. The paper is arranged as follows. In Section 2, we describe briefly the $\frac{G^{\prime }}{G}$ -expansion method. In Sections 3, we apply the method to the (2 + 1)-dimensional dispersive long wave equation. In Section 4 some conclusions are given.

2 Description of the $\frac{G^{\prime }}{G}$ -expansion method

Suppose that a nonlinear equation, say in two independent variables $x\text{,}y$ and t, is given by

step 1: Combining the independent variables x and t into one variable $\xi =x+y+\mathit{wt}$ , we suppose that

step 2: Suppose that the solution of ODE (3) can be expressed by a polynomial in $\frac{G^{\prime }}{G}$ as follows

step 3: By substituting (4) into Eq. (3) and using the second order linear ODE (5), collecting all terms with the same order of $\frac{G^{\prime }}{G}$ together, the left-hand side of Eq. (3) is converted into another polynomial in $\frac{G^{\prime }}{G}$ . Equating each coefficient of this polynomial to zero yields a set of algebraic equations for ${\alpha }_m\text{,}\dots \text{,}\lambda$ and $\mu$ .

step 4: Assuming that the constants ${\alpha }_m\text{,}\dots \text{,}\lambda$ and $\mu$ can be obtained by solving the algebraic equations in step 3, since the general solutions of the second order LODE (5) have been well known for us, then substituting ${\alpha }_m\text{,}\dots \text{,}w$ and the general solutions of Eq. (5) into (4) we have more travelling wave solutions of the nonlinear evolution Eq. (1).

3 (2 + 1)-Dimensional dispersive long wave equation

In this section, we study the following (2 + 1)-DDLW equation in the form

By using (14) and (15) and considering the homogeneous balance between $u^{″}$ and $u^3$ in Eq. (13) we required that $3m=m+2$ then $m=1$ . So we can write (14) as

On solving the Eq. (15), we deduce after some reduction that $$\frac{G^{\prime }}{G}=\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\times \left(\frac{C_1\mathit{sinh}\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_2\mathit{cosh}\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }{C_1\mathit{cosh}\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_2\mathit{sinh}\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }\right)-\frac{\lambda }{2}\text{,}$$ where $C_1$ and $C_2$ are arbitrary constants. Substituting the general solutions of Eq. (15) into (23) we have three types of travelling wave solutions of the DDLW Eqs. (6) and (7) as follows:

When ${\lambda }^2-4\mu >0$

In particular, if $C_1\ne 0\text{,}{C}_2=0\text{,}\lambda >0\text{,}\mu =0$ , u become $$u(\xi )=2\lambda \mathit{tgh}\frac{1}{2}\lambda \xi -\frac{3}{2}\lambda \pm \sqrt{3{\lambda }^2+4\mu +2c+2}\text{.}$$ When ${\lambda }^2-4\mu <0$

And for ${\alpha }_1=-2$ we have

When ${\lambda }^2-4\mu >0$

In particular, if $C_1\ne 0\text{,}{C}_2=0\text{,}\lambda >0\text{,}\mu =0$ , u become $$u(\xi )=2\lambda \mathit{tgh}\frac{1}{2}\lambda \xi +\frac{1}{2}\lambda \pm \sqrt{3{\lambda }^2+4\mu +2c+2}\text{.}$$ When ${\lambda }^2-4\mu <0$

4 Conclusion

In this article we have seen that three types of explicit travelling wave solutions of the (2 + 1)-dimensional dispersive long wave equation are successfully found out by using the $\frac{G^{\prime }}{G}$ -expansion method. The solutions of these nonlinear evolution equations have many potential applications in physics. These equations are very difficult to be solved by traditional methods. The performance of this method is reliable, simple and gives many new exact solutions.