EXACT EXPLICIT TRAVELING WAVE SOLUTIONS FOR A NEW COUPLED ZK SYSTEM

causes the steepening of wave form, whereas the dispersion effect term uxxx in the same equation makes the wave form spread. The balance between this weak nonlinear steepening and dispersion gives rise to solitons. The KdV equation is therefore incapable of shock waves [11]. The KdV equation plays an important role in the development of the soliton theory, where nonlinearity and dispersion dominate, while dissipation effects are small enough to be neglected in the lowest order approximation [1, 2]. The KdV equation is considered a spatially one-dimensional model. An extensive research work has been done in developing higher dimensional models, particularly those in the (2+1), two spatial and one time, dimensions [4]. The best known twodimensional generalizations of the KdV equations are the Kadomtsov-Petviashivilli (KP) equation, and the Zakharov-Kuznetsov (ZK) equation. The ZK equation given by ut + auux + b(uxx + uyy)x = 0, (1.2)


Introduction
The KdV equation is a model that governs the one-dimensional propagation of small-amplitude, weakly dispersive waves [5,6]. The nonlinear term uu x in the KdV equation u t + auu x + u xxx = 0, (1.1) causes the steepening of wave form, whereas the dispersion effect term u xxx in the same equation makes the wave form spread. The balance between this weak nonlinear steepening and dispersion gives rise to solitons. The KdV equation is therefore incapable of shock waves [11]. The KdV equation plays an important role in the development of the soliton theory, where nonlinearity and dispersion dominate, while dissipation effects are small enough to be neglected in the lowest order approximation [1,2]. The KdV equation is considered a spatially one-dimensional model. An extensive research work has been done in developing higher dimensional models, particularly those in the (2 + 1), two spatial and one time, dimensions [4]. The best known twodimensional generalizations of the KdV equations are the Kadomtsov-Petviashivilli (KP) equation, and the Zakharov-Kuznetsov (ZK) equation. The ZK equation given by is investigated in [5,6,10,11,13,14,22] by many distinct approaches. The ZK equation governs the behavior of weakly nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of † the corresponding author. a uniform magnetic field [5,6]. The ZK equation, which is a more isotropic twodimensional, was first derived for describing weakly nonlinear ion-acoustic waves in a strongly magnetized lossless plasma in two-dimensions [22]. It was found that the solitary-wave solutions of the ZK equation are inelastic.
Recently, a new hierarchy of nonlinear evolution equations was derived by Qin [9] by using a finite-dimensional integrable system. An interesting equation in this hierarchy is a new coupled KdV equation where α, β, γ, λ are arbitrary constants. Later, this new coupled equation was investigated by Wu [21], by using matrix transformation and Lax pair. Most recently, in the sense of the KP equation, Wazwaz [15] has extend the new coupled KdV equation to the new coupled KP equation and studied the new coupled KdV equation and the new coupled KP equation, by using the Hirota's bilinear method. The physical phenomena for this system was investigated thoroughly in [9,15,21].
Following the sense of the ZK Eq. (1.2) we can extend the coupled KdV system (1.3) to the new coupled ZK system in the form The derivation of this system is simply made by following the sense of the ZK equation.
Many reliable direct methods are presented to deal with equations arising from physical problems, such as the further improved F-expansion method [18], the multiauxiliary equations expansion method [19], the Riemann-Hilbert method [20], the extended tanh-function method [3], and so on.
In this work, we aim to study the new coupled ZK system. The extended tanhcoth method and the sech method will be mainly used to back up our analysis. The extended tanh-coth method and the sech method are direct and effective algebraic method for handling many nonlinear equations, where solitary wave solutions and triangular periodic solutions are generated.

The methods
In what follows, the methods will be reviewed briefly. Full details can be found in [16,17,12,7,8] and the references therein.
For both methods, we first use the wave variable ξ = x + y − ct to carry a PDE in three independent variables P (u, u t , u x , u y , u xx , u xy , u yy , u xxx , ...) = 0, The extended tanh method admits the use of the finite expansion This will give a system of algebraic equations involving the parameters A k , B k , C k , µ and c. Having determined these parameters, knowing that M is a positive integer in most cases, and using (2.5) we obtain an analytic solution u(x, y, t), v(x, y, t), w(x, y, t) in a closed form.

The sech method
In a manner parallel to the discussion presented above, we use where Z = sechµ(x − ct). The algorithms described above certainly works well for a large class of nonlinear equations. The main advantage of the methods is that it is capable of greatly reducing the size of computational work compared to existing techniques such as the pseudo spectral method, the inverse scattering method, Hirota's bilinear method, and the truncated Painlevé expansion.

Using the tanh-coth method
In this section we employ the extended tanh-coth method to the Eq. (1.4).
Integrating it with ξ and neglecting constants of integration we find Balancing u ′′ with uv in the first equation, v ′′ with wu in the second equation, and w ′′ with uv in the third equation gives The extended tanh-coth method admits the use of the substitution  (1): Case (2): Case (3): Case (4): Case (5): Case (6): Case (7): .
Case (12): Based on these results, we obtain the following solitary solutions for (1.4): (3.15) We can obtain the following triangular periodic solutions for (1.4): and

Discussion
In this paper, we used the extended tanh-coth method and the sech method to study a new coupled ZK equation. As a result, we obtained twenty and four kinds of exact solutions including solitary wave solutions and periodic wave solutions. The methods provided solitary wave solutions and triangular periodic solutions. Moreover, the obtained results in this work clearly demonstrate the reliability of the methods that were used.