The $\left(\frac{G^{\prime }}{G}\right)$ -expansion method for (2 + 1)-dimensional Kadomtsev–Petviashvili equation

1 Introduction

The main idea of this method is that the travelling wave solutions of nonlinear equations can be expressed by a polynomial in $\left(\frac{G^{\prime }}{G}\right)$ where G = G(ξ) satisfies the second order linear ordinary differential equation G″ + λ G′ + μG = 0 where ξ = sx + ly − vt. In recent years, many powerful methods to construct exact solutions of nonlinear evolution equations have been established and developed such as the Jacobi elliptic function expansion, the tanh-method, the truncated Painleve expansion and the $\left(\frac{G^{\prime }}{G}\right)$ -expansion method (Fan, 2000; Inc and Evans, 2004; Liu et al., 2001; Yan, 2003; Yan and Zhang, 1999; Zayed et al., 2005; Zhang et al., 2008; Abdou, 2007; Malfliet, 1992; Parkes and Duffy, 1996; Wang and Li, 2005; Chow, 1995). The rest of the Letter is organized as follows. In Section 2, we describe briefly the $\left(\frac{G^{\prime }}{G}\right)$ -expansion method is briefly described. In Section 3, we apply the method to the (2 + 1)-dimensional Kadomtsev–Petviashvili equation is applied. In Section 4, some conclusions are given.

2 The $\left(\frac{G^{\prime }}{G}\right)$ -expansion method

Now we describe the $\left(\frac{G^{\prime }}{G}\right)$ expansion method for finding travelling, say in three independent variables x, y and t, and is given by

• step 1:

  Combining the independent variables x and t into one variable ξ = x − vt, we suppose that

  (2)

  $$u(x\text{,}t)=u(\xi )\text{,}\xi =\mathit{sx}+\mathit{ly}-\mathit{vt}$$ The travelling wave variable (2) permits us to reduce Eq. (1) to an ODE for G = G(ξ), namely

  (3)

  $$P(u\text{,}{\mathit{su}}^{\prime }\text{,}{\mathit{lu}}^{\prime }\text{,}-{\mathit{vu}}^{\prime }\text{,}{v}^2{u}^{″}\text{,}-{\mathit{vu}}^{″}\text{,}{u}^{″}\text{,}\dots )=0$$

• step 2:

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

  (4)

  $$u(\xi )=\sum _{i=1}^n{\alpha }_i{\left(\frac{G^{\prime }}{G}\right)}^i$$ where G = G(ξ) satisfies the second order LODE in the form

  (5)

  $$G^{″}+\lambda G^{\prime }+\mu G=0$$ α_n,…,λ and μ are constants to be determined later α_n ≠ 0. The positive integer “n” can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in (3)

• step 3:

  By substituting (4) into Eq. (3) and using the second order linear ODE (5), collecting all terms with the same order $\left(\frac{G^{\prime }}{G}\right)$ together, the left-hand side of Eq. (3) is converted into another polynomial in $\left(\frac{G^{\prime }}{G}\right)$ . Equating each coefficient of this polynomial to zero yields a set of algebraic equations for α_n,…,λ and μ. By solving the algebraic equations above we obtain α_n, … , v.

3 (2 + 1)-Dimensional Kadomtsev–Petviashvili equation

We consider the (2 + 1)-dimensional Kadomtsev–Petviashvili equation in the form

By using (9) and (10) and considering the homogeneous balance between u″ and u² in Eq. (8) we required that 2n = n + 2 then n = 2. So we can write (9) as

By using (13), expression (11) can be written as

• Case 1:

  When λ² − 4μ ≻ 0 $$u(\xi )=-12{\epsilon }^2{s}^2({\lambda }^2-4\mu )\times \left(\frac{C_1\sinh \frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_2\cosh \frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }{C_1\cosh \frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +C_2\sinh \frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi }\right)-12{\epsilon }^2{s}^2\lambda -\frac{\lambda }{2}+{\alpha }_0$$ where $\xi =x-\frac{1}{s}(8{\epsilon }^2{s}^2+{\epsilon }^2{s}^4{\lambda }^2+s^2{\alpha }_0+l^2\delta t$ . C₁, and C₂, are arbitrary constants.

  If C₁ and C₂ are taken as special values, the various known results in the literature can be rediscovered, for instance, if $C_1\succ 0\text{,}{C}_1^2\succ C_2^2\text{,}$ then u = u(ξ) can be written as $$u(\xi )=-12{\epsilon }^2{s}^2({\lambda }^2-4\mu )\times \sec h^2\left(\frac{1}{2}\sqrt{{\lambda }^2-4\mu }\xi +{\xi }_0\right)-12{\epsilon }^2{s}^2\lambda -\frac{\lambda }{2}+{\alpha }_0$$

• Case 2:

  When ${\lambda }^2-4\mu \prec 0$ $$u(\xi )=-12{\epsilon }^2{s}^2({\lambda }^2-4\mu )\times \left(\frac{-C_1\sin \frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi +C_2\cos \frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }{C_1\cos \frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi +C_2\sin \frac{1}{2}\sqrt{4\mu -{\lambda }^2}\xi }\right)-12{\epsilon }^2{s}^2\lambda -\frac{\lambda }{2}$$

• Case 3:

  When λ² − 4μ = 0 $$u(\xi )=\frac{-12{\epsilon }^2{s}^2{C}_2}{{(C_1+C_2\xi )}^2}-12{\epsilon }^2{s}^2\lambda -\frac{\lambda }{2}+{\alpha }_0$$ where C₁ and C₂ are arbitrary constants.

4 Conclusions

The solutions of these nonlinear evolution equations have many potential applications in physics. In this paper, we have seen that three types of travelling solutions of (2 + 1)-dimensional Kadomtsev–Petviashvili equation are successfully found out by using the $\left(\frac{G^{\prime }}{G}\right)$ -expansion method. The performance of this method is reliable, simple and gives many new exact solutions.