Research Article  Open Access
R. Naz, Z. Ali, I. Naeem, "Reductions and New Exact Solutions of ZK, Gardner KP, and Modified KP Equations via Generalized Double Reduction Theorem", Abstract and Applied Analysis, vol. 2013, Article ID 340564, 11 pages, 2013. https://doi.org/10.1155/2013/340564
Reductions and New Exact Solutions of ZK, Gardner KP, and Modified KP Equations via Generalized Double Reduction Theorem
Abstract
We study here the Lie symmetries, conservation laws, reductions, and new exact solutions of () dimensional ZakharovKuznetsov (ZK), Gardner KadomtsevPetviashvili (GKP), and Modified KadomtsevPetviashvili (MKP) equations. The multiplier approach yields three conservation laws for ZK equation. We find the Lie symmetries associated with the conserved vectors, and three different cases arise. The generalized double reduction theorem is then applied to reduce the thirdorder ZK equation to a secondorder ordinary differential equation (ODE) and implicit solutions are established. We use the SineCosine method for the reduced secondorder ODE to obtain new explicit solutions of ZK equation. The Lie symmetries, conservation laws, reductions, and exact solutions via generalized double reduction theorem are computed for the GKP and MKP equations. Moreover, for the GKP equation, some new explicit solutions are constructed by applying the first integral method to the reduced equations.
1. Introduction
The association of conservation laws with Noether symmetries [1], LieBäcklund symmetries [2], and nonlocal symmetries [3, 4] has been of great interest during the last few decades. This association results in double reduction of a partial differential equation (PDE). For variational partial differential equations (PDEs), the double reduction was achieved by association of a Noether symmetry with a conserved vector [5, 6]. Sjöberg [7, 8] developed a double reduction formula for a nonvariational PDE of order with two independent and dependent variables to reduce it to an ODE of order provided that the PDE admits a nontrivial conserved vector associated with at least one symmetry. Recently, Bokhari et al. [9] generalized the double reduction theory for the case of several independent variables. According to the generalized double reduction theory, a nonlinear system of thorder PDEs with independent and dependent variables can be reduced to a nonlinear system of thorder ODEs. In every reduction, at least one symmetry should be associated with a nontrivial conserved vector; otherwise, reduction is not possible. Naz et al. [10] utilized the double reduction theory to find some exact solutions of a class of nonlinear regularized long wave equations.
Different methods are developed for the construction of conservation laws compared by Naz et al. [11], and see also references therein. We will use the multiplier approach. The conservation law in characteristic form [12] can be expressed as , and one can compute the characteristics (multipliers) by taking the variational derivative of for the arbitrary functions not only for solutions of system of partial differential equations [6]. It was successfully applied to construct the conservation laws (see, e.g., [11, 13]).
In this paper, we consider dimensional ZK [14, 15], GKP [16], and MKP [17] equations. The conservation laws are computed by the multiplier approach. The symmetry conservation law relation is used to determine symmetries associated with the conserved vectors. Reductions and new exact solutions are found by the generalized double reduction theory for ZK, GKP, and MKP equations. We utilize the SineCosine method [18–20] and first integral method [21] to compute new explicit solutions for the reduced conserved forms of ZK and GKP equations. To the best of our knowledge, the exact solutions derived here are new and not reported in the literature.
The detail outline of the paper is as follows. In Section 2, basic definitions, important relations, and the fundamental theorem of generalized double reduction theory are presented. The Lie symmetries, conservation laws, reduced forms, and new exact solutions via generalized double reduction theorem for ZK equation are constructed in Section 3. In Sections 4 and 5, Lie symmetries, conservation laws, reductions, and new exact solutions of GKP and MKP equations are studied. Concluding remarks are summarized in Section 6.
2. Fundamental Operators
The following definitions are adopted from the literature [7–9, 11, 22].
Consider the following thorder system of PDEs: where are the independent variables, and are the dependent variables.
Definition 1. A LieBäcklund or generalized operator is defined by and the additional coefficients can be found from in which is the Lie characteristic function described by
Definition 2. The Euler operator is defined by where is the total derivative operator with respect to .
Definition 3. A conserved vector , , satisfies for all solutions of (1) is called a local conservation law. Here denotes the space of all differential functions.
Definition 4. A LieBäcklund operator given in (2) is associated with the conserved vector of (1) if it satisfies the following relation:
Equation (7) is known as the symmetry conservation laws relationship [22].
New conservation laws can be derived from existing conservation laws and the symmetries by using the following theorem adopted from [22, 23].
Theorem 5. Suppose is any LieBäcklund operator of (1) and comprise the components of a conserved vector of (1) then yields the components of a conserved vector of (1), and thus
Theorem 6 (see [9]). Suppose is a conservation law of the PDE system (1). Then under a contact transformation, there exist functions such that where is given by In (10), , , and can be determined from and .
The following is the fundamental theorem on double reduction theory [9].
Theorem 7. Suppose is a conservation law of the PDE system (1). Then under a similarity transformation of a symmetry of the form (2) for the PDE, there exist functions such that is still symmetry for the PDE and where
Corollary 8 (the necessary and sufficient condition for reduced conserved form [9]). The conserved form of the PDE system (1) can be reduced under a similarity transformation of a symmetry to a reduced conserved form if and only if is associated with the conservation law , that is, .
Corollary 9 (see [9]). A nonlinear system of thorder PDEs with independent and dependent variables which admits a nontrivial conserved form that has at least one associated symmetry in every reduction from the reductions (the first step of double reduction) can be reduced to a thorder nonlinear system of ODEs.
3. Lie Symmetries, Conservation Laws, Reductions, and New Exact Solutions of ZakharovKuznetsov Equation
The () dimensional ZakharovKuznetsov (ZK) equation [14, 15] representing the model for nonlinear Rossby waves is where , , , and are arbitrary constants. First we will derive the Lie symmetries of (14). The Lie point symmetry generator of (14), is derived by solving in which is the third prolongation and can be computed from (2). Equation (16), after expansion and separation, yields the following overdetermined system of partial differential equations for the unknown coefficients , , , and : The solution of system (17) gives the following Lie symmetries: The conservation laws for (14) will be derived by the multiplier approach. Consider the multipliers of the form for (14). The determining equation for the multipliers is where is the standard Euler operator and can be computed from (5). Expanding and then separating (19) with respect to different combinations of derivatives of yields the following overdetermined system for the multipliers: The solution of system (20) can be expressed as where and are arbitrary constants, and is an arbitrary function of . Multipliers for (14) satisfy for all functions . From (21) and (22), we obtain three conserved vectors, and they are given in Table 1. If we consider multipliers of form , we get the same conserved vectors.

Now, we apply the double reduction theorem based on Lie symmetries and conservation laws to find the reductions and exact solutions. Let , , and . Equation (7) for variables , , and yields Equation (23) is used to find symmetries associated with the conserved vectors presented in Table 1. The symmetries , , and are associated with the conserved vector . The symmetries , , , and are associated with the conserved vector only when .
3.1. Reduction via Using Combination of Symmetries
The conserved vector for yields The combination of symmetries , , and can be used to obtain a reduced conserved form. The generator, , has a canonical form when and thus the canonical variables are The formula (10) for the reduced conserved form in terms of variables and can be expressed as where from (11) is given by Equations (28) and (29) for the conserved vector (24) results in and reduced conserved form is The generalized double reduction theorem reduced the thirdorder ZK equation (14) from the thirdorder PDE in terms of three independent variables to a system of two secondorder PDES with two independent variables . It can be further reduced to an ODE if the reduced form admits symmetries, and at least one symmetry is associated with a nontrivial conserved vector. The reduced conserved form (31) admits the following two symmetries: Since and satisfy the symmetry conservation law relation therefore , and are the associated symmetries, and it is possible to find second reduction. A reduced conserved form can be obtained using The canonical form of generator is with the similarity variables In this case, the formula (10) for the reduced conserved form results in with Equations (36) and (37) for the conserved vector (30) take the following form: Equation (38) expressed in terms of variable becomes and the reduced conserved form is Equation (40) gives , and (39) can be written as The symmetries , , and are associated with the conserved vector only when . For this case, the generalized double reduction theorem is applied twice to ZK equation (14) and it is reduced to secondorder ODE (41). Next, we find implicit and explicit solutions of reduced form (41) and these constitute the exact solutions of ZK equation (14).
The implicit solution of (41) using Maple isNow, we compute the explicit solutions of (41) by utilizing the SineCosine method [18–20]. The solution of (41) can be expressed in the form or where , and are parameters need to be determined.
Substituting the values of from (43) and setting in (41) yields where and . Equation (45) is satisfied if Ultimately, the solution of algebraic system (46) yields the solution of (41) and is given by Similarly, using the Sine function (44) one can easily obtain the solution of (41) as where , . The solutions (47) and (48) can be finally expressed in terms of original variables as and these constitute the exact solutions of ZK equation (14).
3.2. Reduction via Using Combination of Symmetries , , and
The symmetries ,, and are associated with the conserved vector , and a reduced conserved form can be obtained by the combination given in (25). For the conserved vector in terms of canonical variables (27), we have and reduced conserved form is . The reduced conserved form admits the symmetries (32) and these symmetries satisfy the symmetry conservation law relation (33) for the conserved vectors and . The combination of symmetries yields the generator and the similarity variables (35). Equations (36) and (37) for the conserved vector (51) gives The reduced conserved form is and this yields where is a constant. Using Maple, the solution of (54) iswhereas (54) gives the same solution by employing SineCosine method as we have obtained in the previous case. The symmetries , , and are associated with the conserved vector , and in this case generalized double reduction theorem gives one implicit solution (55) for the ZK equation (14). It is interesting to notice that generalized double reduction theorem yields two different reduced forms (41) and (54) for traveling wave solutions (49) and (50), whereas in [14, 15] only one reduced form (41) was obtained. One can use the simple reduced form to construct exact or approximate solutions.
3.3. Reduction via Using Symmetry
The generator, , has canonical form if and thus we have Equations (28) and (29) for the conserved vector in terms of canonical variables (57) result in and reduced conserved form is . The conserved form (58) cannot be further reduced because it does not admit any symmetry, however one can perform the numerical simulation or any other approximate method to construct the approximate solutions.
The generalized double reduction theorem gives two different reduced forms (41) and (54) for traveling wave solutions. The SineCosine method for each of the reduced forms gives the explicit solutions (49) and (50) for the ZK equation. Also we find two implicit solutions (42) and (55) by Maple. The transformations (57) are obtained due to double reduction theorem, and these transformations are different from the traveling wave transformations. These transformations provide the reduced form (58), and numerical method can be applied to obtain approximate solutions for the ZK equation (14). The exact solutions for ZK equation obtained here are different from the class of exact solutions computed by Expfunction method [14] and by transformation of elliptic equation [15].
4. Lie Symmetries, Conservation Laws, and Exact Solutions of Gardner KP Equation
The Gardner KP equation [16] is The Lie symmetry generator determining equation for Gardner KP equation (59) is where is the fourth prolongation. Solving (60), after expansion, we have as the Lie symmetry generators for the Gardner KP equation (59). Consider the multipliers of the form for (59). The determining equation for the multipliers is Equation (62) finally presents and this results in four multipliers. The multipliers and conserved vectors are presented in Table 2.

The symmetry conservation law relation (23) is not satisfied for the conserved vectors , , and . The symmetries , , and are associated with the conserved vector if . Thus we can get a reduced conserved form by the combination of given in (25). Equations (28) and (29) for the conserved vector in terms of canonical variables (27) yield the following three components of : and reduced conserved form is The reduced conservation law admits the following symmetries: The symmetries and satisfy the symmetry conservation law relation (33) for the conserved vectors and . Taking the combination of these symmetries yields the same generator as given in (34), and the canonical form can be obtained from similarity variables (35). Using formula (36), one has the following two components of : The reduced conserved form satisfies , and we have The integration of (68) provides Next, we find implicit and explicit solutions of reduced form (69) and these constitute the exact solutions of GKP equation (59).
Equation (69) gives the following solution if : where , and , and are constants.
For explicit solution, we apply the first integral method to the reduced form (69). We substitute and with which converts (69) into the following system of ODEs: where . Next, we apply the division theorem to seek the first integral to (71). Assume that and are the nontrivial solutions to (71) and is an irreducible polynomial in such that where , are polynomials of and all relatively prime in , . Equation (73) is also called the first integral to (71). Suppose that in (73). By division theorem, there exist polynomials in such that or Substituting from (71) in (75) and then separating with respect to powers of , we obtain Solving the system (76) for and and then substituting it into (73) yields an ODE which finally gives The explicit solutions (77) form exact solution of GKP equation (59).
The generalized double reduction theorem is applied twice to the GKP equation (59), and it is reduced to an integrable thirdorder ODE (68). On integration, the third order ODE (68) is further reduced to secondorder ODE (69). Using Maple equation (69) yields one implicit solution (70) for the GKP equation (59). Also two explicit solutions (77) for the GKP equation are obtained utilizing the first integral method to the reduced secondorder ODE (69). The exact solutions derived here are different from class of multiplesoliton solutions obtained by Hirota’s bilinear method [16].
5. Lie Symmetries, Conservation Laws, and Exact Solutions of Modified KP Equation
The MKP equation [17] describing the soliton propagation in multitemperature electrons plasmas is where , , , and are plasma parameters. For (78) the multipliers of the form are considered. The multipliers and conserved vectors are given in Table 3. The MKP equation (78) has the following Lie symmetry generators: Only the Lie symmetries , , and are associated with the conserved vector when . By using their combination as we have done in Section 3, and with the aid of , we obtain A particular solution of (80) can be found for the case and is given by where , .
