Symmetry Reductions and Exact Solutions of a Variable Coefficient (2+1)-zakharov-kuznetsov Equation

We study the generalized (2+1)-Zakharov-Kuznetsov (ZK) equation of time dependent variable coefficients from the Lie group-theoretic point of view. The Lie point symmetry generators of a special form of the class of equations are derived. We classify the Lie point symmetry generators to obtain the optimal system of one-dimensional subalgebras of the Lie symmetry algebras. These subalgebras are then used to construct a number of symmetry reductions and exact group-invariant solutions to the underlying equation.


INTRODUCTION
The study of the exact solutions of nonlinear evolution equations plays an important role to understand the nonlinear physical phenomena which are described by these equations.The importance of deriving such exact solutions to these nonlinear equations facilitate the verification of numerical methods and helps in the stability analysis of solutions.
In this paper, we study the exact solutions of one such nonlinear evolution equation, the generalized (2+1)-Zakharov-Kuznetsov equation of the form 0 of time dependent variable coefficients.Here f(t), g(t) and h(t) are arbitrary smooth functions of the variable t and 0  fgh .The equation (1) models the nonlinear development of ion-acoustic waves in a magnetized plasma under the restrictions of small wave amplitude, weak dispersion, and strong magnetic fields [1].The equation (1) also appears in different forms in many areas of Physics, Applied Mathematics and Engineering (see for example [2,3]).
in our analysis as all the results of the class (4) can be extended to the class (1) by the transformation (2).
In [4], travelling wave-like solutions for the equation (1) were obtained.In [5] and [6], similarity reductions and some exact solutions were obtained for the special cases of the class of equations ( 4) using symmetry group method.For the theory and application of the Lie symmetry methods, see e.g., the Refs.[7,8,9,10].Recently, in [11] the method of Lie groups is utilized to derive solutions to an integrable equation governing short waves in a long-wave model.The outline of the paper is as follows.In Section 2, we present the Lie point symmetries of a special case of the equation ( 4).In Section 3, we construct the optimal system of one-dimensional subalgebras of the Lie symmetry algebra of the special form of the equation.Moreover, using the optimal system of subalgebras symmetry reductions and exact group-invariant solutions of the underlying equation are obtained.Finally, in Section 4 concluding remarks are made.

LIE POINT SYMMETRIES
In this section, we consider a special case of the class of equations (4 is a generator of point symmetry of the equation (5 where the operator ] 3 [ X is the third prolongation of the operator X defined by and xyy  are given by Here i D denotes the total derivative operator and is defined by The coefficient functions , and  are calculated by solving the determining equation (7).Since    , , and  are independent of the derivatives of u , the coefficients of like derivatives of u in ( 7) can be equated to yield an over determined system of linear partial differential equations (PDEs).Therefore, the determining equation for symmetries after lengthy calculations yield Solving the determining equations ( 8)-(13) for    , , and  , we obtain the following symmetry group generators given by ., , ,

SYMMETRY REDUCTIONS AND EXACT GROUP-INVARIANT SOLUTIONS OF THE EQUATION (5)
Here we first construct the optimal system of one-dimensional subalgebras of the Lie algebra admitted by the equation (5).The classification of the one-dimensional subalgebras are then used to reduce the equation ( 5) into a partial differential equation (PDE) having two independent variables.Then we also study the symmetry properties of the reduced PDE to derive further symmetry reductions and exact group-invariant solutions for the underlying equation.
The results on the classification of the Lie point symmetries of the equation ( 5) are summarized by the Tables 1, 2 and 3.The commutator table of the Lie point symmetries of the equation ( 5) and the adjoint representations of the symmetry group of (5) on its Lie algebra are given in Table 1 and Table 2, respectively.The Table 1 and Table 2 are used to construct the optimal system of one-dimensional subalgebras for equation (5) which is given in Table 3 (for more details of the approach see [8] and the references therein).
Table 1.Commutator table of the Lie algebra of equation ( 5) X Table 3. Subalgebra, group invariants, group-invariant solutions of (5) N X and  , and  are arbitrary constants.
Case 1.In this case, the group-invariant solution corresponding to the symmetry generator Now the equation ( 14) admits the following symmetry generators given by ., , where X , the substitution of this solution into the equation ( 14) and solving we obtain a solution , where is the group invariant.Substitution of this solution into the equation ( 14) gives rise to the ordinary differential equation (ODE here `prime' denotes differentiation with respect to  .

Case 2. The group-invariant solution arising from
The equation ( 16) admits the following three Lie point symmetry generators ., , The optimal system of one-dimensional subalgebras are , , , where c is an arbitrary real constant and .
The group-invariant solution corresponding to , where , the substitution of this solution into the equation ( 16) results in the following ODE , 0 ) ( here `prime' denotes differentiation with respect to . , where    is the group invariant.Substitution of this solution into the equation ( 16) gives the solution The symmetry generator 1 X gives the trivial solution , where C is a constant.
Case 3. The group-invariant solution that corresponds to Hence the solution of the equation ( 5) is given by , The equation (20) admits the Lie algebra spanned by the following symmetry generators , where X , the substitution of this solution into the equation ( 20) and solving we obtain the solution t , where    is the group invariant of 1 X , the substitution of this solution into the equation ( 19) and solving we obtain the solution , where C is a constant. (ii) where    is the group invariant of

CONCLUDING REMARKS
In this paper we have studied the generalized (2+1)-ZK equation with time dependent variable coefficients using the Lie symmetry group method.We derived the Lie point symmetry generators of a special form of the underlying class of equations.The Lie symmetry classification with respect to the special form of the time dependent variable coefficients equation was presented.We used this classification of optimal system of one-dimensional subalgebras of the Lie symmetry algebras to construct symmetry reductions and exact group-invariant solutions for the special form of the equation.
substitution of this solution into the equation (20) and solving we obtain the solution , instance, the PDE (19) admits the following symmetry generators .