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 (2+1) dimensional Zakharov-Kuznetsov (ZK), Gardner Kadomtsev-Petviashvili (GKP), and Modified Kadomtsev-Petviashvili (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 third-order ZK equation to a second-order ordinary differential equation (ODE) and implicit solutions are established. We use the Sine-Cosine method for the reduced second-order 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], Lie-Bäcklund 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 𝑞 with two independent and 𝑚 dependent variables to reduce it to an ODE of order (𝑞−1) 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 𝑞th-order PDEs with 𝑛 independent and 𝑚 dependent variables can be reduced to a nonlinear system of (𝑞−1)th-order 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 𝐷_𝑖𝑇^𝑖=Λ^𝛼𝐸_𝛼, and one can compute the characteristics (multipliers) by taking the variational derivative of 𝐷_𝑖𝑇^𝑖=𝑄^𝛼𝐸_𝛼 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 (2+1) 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 Sine-Cosine 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 𝑞th-order system of PDEs: 𝐸^𝜈=𝑥,𝑢,𝑢₍₁₎,𝑢₍₂₎,…,𝑢_(𝑞),𝜈=1,2,3,…,𝑚, (1) where 𝑥=(𝑥¹,𝑥²,𝑥³,…,𝑥^𝑛) are the independent variables, and 𝑢=(𝑢¹,𝑢²,𝑢³,…,𝑢^𝑚) are the dependent variables.

Definition 1. A Lie-Bäcklund or generalized operator is defined by 𝑋=𝜉^𝑖𝜕𝜕𝑥^𝑖+𝜂^𝛼𝜕𝜕𝑢^𝛼+𝜁^𝛼_𝑖𝜕𝜕𝑢^𝛼_𝑖+_𝑠≥1𝜁^𝛼_{𝑖1⋯𝑖𝑠}𝜕𝜕𝑢^𝛼_{𝑖1⋯𝑖𝑠}, (2) and the additional coefficients 𝜁^𝛼_{𝑖1⋯𝑖𝑠} can be found from 𝜁^𝛼_𝑖=𝐷_𝑖𝑊^𝛼+𝜉^𝑗𝑢^𝛼_𝑖𝑗,𝜁^𝛼_{𝑖1⋯𝑖𝑠}=𝐷_𝑖1⋯𝐷_𝑖𝑠𝑊^𝛼+𝜉^𝑗𝑢^𝛼_{𝑗𝑖1⋯𝑖𝑠},𝑠>1, (3) in which 𝑊^𝛼 is the Lie characteristic function described by 𝑊^𝛼=𝜂^𝛼−𝜉^𝑗𝑢^𝛼_𝑗. (4)

Definition 2. The Euler operator is defined by 𝛿𝛿𝑢^𝛼=𝜕𝜕𝑢^𝛼−𝐷_𝑖𝜕𝜕𝑢^𝛼_𝑖+𝐷_𝑖𝐷_𝑗𝜕𝜕𝑢^𝛼_𝑖𝑗−⋯, (5) where 𝐷_𝑖=𝜕𝜕𝑥^𝑖+𝑢^𝛼_𝑖𝜕𝜕𝑢^𝛼+𝑢^𝛼_𝑖𝑗𝜕𝜕𝑢^𝛼_𝑗+⋯,𝑖=1,2,…,𝑛 (6) is the total derivative operator with respect to 𝑥^𝑖.

Definition 3. A conserved vector 𝑇=(𝑇¹,𝑇²,…,𝑇^𝑛), 𝑇^𝑖∈𝒜, 𝑖=1,2,…,𝑛 satisfies 𝐷_𝑖𝑇^𝑖|₍₁₎=0 for all solutions of (1) is called a local conservation law. Here 𝒜 denotes the space of all differential functions.

Definition 4. A Lie-Bäcklund operator 𝑋 given in (2) is associated with the conserved vector 𝑇 of (1) if it satisfies the following relation: X𝑇^𝑖+𝑇^𝑖𝐷_𝑗𝜉^𝑗−𝑇^𝑗𝐷_𝑗𝜉^𝑖=0,𝑖=1,2,…,𝑛. (7) 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 𝑋 is any Lie-Bäcklund operator of (1) and 𝑇^𝑖,𝑖=1,2,3,…,𝑛 comprise the components of a conserved vector of (1) then 𝑇^𝑖=𝑋𝑇^𝑖+𝑇^𝑖𝐷_𝑗𝜉^𝑗−𝑇^𝑗𝐷_𝑗𝜉^𝑖,𝑖=1,2,3,…,𝑛 (8) yields the components of a conserved vector of (1), and thus 𝐷_𝑖𝑇^𝑖₍₁₎=0. (9)

Theorem 6 (see [9]). Suppose 𝐷_𝑖𝑇^𝑖=0 is a conservation law of the PDE system (1). Then under a contact transformation, there exist functions 𝑇^𝑖 such that 𝐽𝐷_𝑖𝑇^𝑖=𝐷_𝑖𝑇^𝑖 where 𝑇^𝑖 is given by 𝑇¹𝑇²⋅⋅⋅𝑇^𝑛=𝐽𝐴^−1𝑇𝑇¹𝑇²⋅⋅⋅𝑇^𝑛,𝐽𝑇¹𝑇²⋅⋅⋅𝑇^𝑛=𝐴^𝑇𝑇¹𝑇²⋅⋅⋅𝑇^𝑛. (10) In (10), 𝐴, 𝐴⁻¹, and 𝐽 can be determined from 𝐷𝐴=¹𝑥₁𝐷₁𝑥₂⋅⋅⋅𝐷₁𝑥_𝑛𝐷₂𝑥₁𝐷₂𝑥₂⋅⋅⋅𝐷₂𝑥_𝑛𝐷⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅_𝑛𝑥₁𝐷_𝑛𝑥₂⋅⋅⋅𝐷_𝑛𝑥_𝑛,𝐴⁻¹=𝐷₁𝑥₁𝐷₁𝑥₂⋅⋅⋅𝐷₁𝑥_𝑛𝐷₂𝑥₁𝐷₂𝑥₂⋅⋅⋅𝐷₂𝑥_𝑛𝐷⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅_𝑛𝑥₁𝐷_𝑛𝑥₂⋅⋅⋅𝐷_𝑛𝑥_𝑛, (11) and 𝐽=det(𝐴).

The following is the fundamental theorem on double reduction theory [9].

Theorem 7. Suppose 𝐷_𝑖𝑇^𝑖=0 is a conservation law of the PDE system (1). Then under a similarity transformation of a symmetry 𝑋 of the form (2) for the PDE, there exist functions 𝑇^𝑖 such that 𝑋 is still symmetry for the PDE 𝐷_𝑖𝑇^𝑖=0 and 𝑋𝑇¹𝑋𝑇²⋅⋅⋅𝑋𝑇^𝑛=𝐽𝐴^−1𝑇𝑇¹𝑇,𝑋₂⋅⋅⋅𝑇,𝑋_𝑛,𝑋, (12) where 𝐷𝐴=¹𝑥₁𝐷₁𝑥₂⋅⋅⋅𝐷₁𝑥_𝑛𝐷₂𝑥₁𝐷₂𝑥₂⋅⋅⋅𝐷₂𝑥_𝑛𝐷⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅_𝑛𝑥₁𝐷_𝑛𝑥₂⋅⋅⋅𝐷_𝑛𝑥_𝑛,𝐴⁻¹=𝐷₁𝑥₁𝐷₁𝑥₂⋅⋅⋅𝐷₁𝑥_𝑛𝐷₂𝑥₁𝐷₂𝑥₂⋅⋅⋅𝐷₂𝑥_𝑛𝐷⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅_𝑛𝑥₁𝐷_𝑛𝑥₂⋅⋅⋅𝐷_𝑛𝑥_𝑛,𝐽=det(𝐴). (13)

Corollary 8 (the necessary and sufficient condition for reduced conserved form [9]). The conserved form 𝐷_𝑖𝑇^𝑖=0 of the PDE system (1) can be reduced under a similarity transformation of a symmetry 𝑋 to a reduced conserved form 𝐷_𝑖𝑇^𝑖=0 if and only if 𝑋 is associated with the conservation law 𝑇, that is, [𝑇,𝑋]|₍₁₎=0.

Corollary 9 (see [9]). A nonlinear system of 𝑞th-order PDEs with 𝑛 independent and 𝑚 dependent variables which admits a nontrivial conserved form that has at least one associated symmetry in every reduction from the 𝑛 reductions (the first step of double reduction) can be reduced to a (𝑞−1)th-order nonlinear system of ODEs.

3. Lie Symmetries, Conservation Laws, Reductions, and New Exact Solutions of Zakharov-Kuznetsov Equation

The (2+1) dimensional Zakharov-Kuznetsov (ZK) equation [14, 15] representing the model for nonlinear Rossby waves is 𝑢_𝑡+𝑎𝑢_𝑥+𝑏𝑢𝑢_𝑥+𝑐𝑢_𝑥𝑥𝑥+𝑑𝑢_𝑥𝑦𝑦=0, (14) where 𝑎, 𝑏, 𝑐, and 𝑑 are arbitrary constants. First we will derive the Lie symmetries of (14). The Lie point symmetry generator 𝑋=𝜉¹𝜕𝑡,𝑥,𝑦,𝑢𝜕𝑡+𝜉²𝜕𝑡,𝑥,𝑦,𝑢𝜕𝑥+𝜉³𝜕𝑡,𝑥,𝑦,𝑢𝜕𝜕𝑦+𝜂𝑡,𝑥,𝑦,𝑢,𝜕𝑢 (15) of (14), is derived by solving 𝑋^[3]𝑢_𝑡+𝑎𝑢_𝑥+𝑏𝑢𝑢_𝑥+𝑐𝑢_𝑥𝑥𝑥+𝑑𝑢_{𝑥𝑦𝑦(14)}=0, (16) in which 𝑋^[3] 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 𝜉¹, 𝜉², 𝜉³, and 𝜂: 𝜉¹_𝑢=0,𝜉¹_𝑥=0,𝜉¹_𝑦𝜉=0,_2𝑢=0,𝜉²_𝑦=0,𝜉²_𝑥𝑥𝜉=0,_3𝑢=0,𝜉³_𝑡=0,𝜉³_𝑥𝜂=0,_𝑢𝑢=0,𝜂_𝑥𝑢=0,𝜉³_𝑦−𝜉²_𝑥𝜉=0,_3𝑦𝑦−2𝜂_𝑦𝑢=0,𝜉¹_𝑡−3𝜉²_𝑥𝜂=0,_𝑡+(𝑎+𝑏𝑢)𝜂_𝑥+𝑐𝜂_𝑥𝑥𝑥+𝑑𝜂_𝑥𝑦𝑦𝜉=0,_2𝑡−2(𝑎+𝑏𝑢)𝜉²_𝑥−𝑏𝜂−𝑑𝜂_𝑦𝑦𝑢=0. (17) The solution of system (17) gives the following Lie symmetries: 𝑋₁=𝜕𝜕𝑡,𝑋₂=𝜕𝜕𝑥,𝑋₃=𝜕,𝑋𝜕𝑦₄=𝜕𝜕𝜕𝑢+𝑏𝑡,𝑋𝜕𝑥₅𝜕=3𝑡+()𝜕𝜕𝑡𝑥+2𝑎𝑡𝜕𝜕𝑥+𝑦𝜕𝜕𝑦−2𝑢.𝜕𝑢 (18) The conservation laws for (14) will be derived by the multiplier approach. Consider the multipliers of the form Λ=Λ(𝑡,𝑥,𝑢) for (14). The determining equation for the multipliers is 𝛿Λ𝑢𝛿𝑢^𝑡+𝑎𝑢_𝑥+𝑏𝑢𝑢_𝑥+𝑐𝑢_𝑥𝑥𝑥+𝑑𝑢_𝑥𝑦𝑦=0, (19) where 𝛿/𝛿𝑢 is the standard Euler operator and can be computed from (5). Expanding and then separating (19) with respect to different combinations of derivatives of 𝑢 yields the following overdetermined system for the multipliers: Λ_𝑥𝑥=0,Λ_𝑥𝑦=0,Λ_𝑢𝑥Λ=0,_𝑢𝑦=0,Λ_𝑢𝑢Λ=0,_𝑡=−Λ_𝑥(𝑏𝑢+𝑎),Λ_𝑢𝑡=0. (20) The solution of system (20) can be expressed as Λ=𝑓𝑦+(𝑥−𝑎𝑡−𝑏𝑡𝑢)𝑐₁+𝑐₂𝑢, (21) where 𝑐₁ and 𝑐₂ are arbitrary constants, and 𝑓(𝑦) is an arbitrary function of 𝑦. Multipliers Λ for (14) satisfy Λ𝑢_𝑡+𝑎𝑢_𝑥+𝑏𝑢𝑢_𝑥+𝑐𝑢_𝑥𝑥𝑥+𝑑𝑢_𝑥𝑦𝑦=𝐷_𝑡𝑇^𝑡+𝐷_𝑥𝑇^𝑥+𝐷_𝑦𝑇^𝑦, (22) for all functions 𝑢(𝑡,𝑥,𝑦). From (21) and (22), we obtain three conserved vectors, and they are given in Table 1. If we consider multipliers of form Λ=Λ(𝑡,𝑥,𝑢,𝑢_𝑡,𝑢_𝑥,𝑢_𝑦), we get the same conserved vectors.

Table 1 

Multipliers and conserved vectors for (14).

Now, we apply the double reduction theorem based on Lie symmetries and conservation laws to find the reductions and exact solutions. Let 𝑇₁=(𝑇^𝑡₁,𝑇^𝑥₁,𝑇^𝑦₁), 𝑇₂=(𝑇^𝑡₂,𝑇^𝑥₂,𝑇^𝑦₂), and 𝑇₃=(𝑇^𝑡₃,𝑇^𝑥₃,𝑇^𝑦₃). Equation (7) for variables 𝑡, 𝑥, and 𝑦 yields 𝑋𝑇^𝑡𝑇^𝑥𝑇^𝑦−𝐷_𝑡𝜉¹𝐷_𝑥𝜉¹𝐷_𝑦𝜉¹𝐷_𝑡𝜉²𝐷_𝑥𝜉²𝐷_𝑦𝜉²𝐷_𝑡𝜉³𝐷_𝑥𝜉³𝐷_𝑦𝜉³𝑇^𝑡𝑇^𝑥𝑇^𝑦+𝐷_𝑡𝜉¹+𝐷_𝑥𝜉²+𝐷_𝑦𝜉³𝑇^𝑡𝑇^𝑥𝑇^𝑦=0. (23) Equation (23) is used to find symmetries associated with the conserved vectors presented in Table 1. The symmetries 𝑋₁,   𝑋₂,  and  𝑋₃ are associated with the conserved vector 𝑇₂. The symmetries 𝑋₁,  𝑋₂,  𝑋₃, and 𝑋₅ are associated with the conserved vector 𝑇₃ only when 𝑓(𝑦)=1.

3.1. Reduction via 𝑇₃ Using Combination of Symmetries 𝑋₁,𝑋₂,𝑋₃

The conserved vector 𝑇₃ for 𝑓(𝑦)=1 yields 𝑇^𝑡₃=𝑢,𝑇^𝑥₃=𝑢²𝑏2+𝑎𝑢+𝑐𝑢_𝑥𝑥,𝑇^𝑦₃=𝑑𝑢_𝑥𝑦. (24) The combination of symmetries 𝑋₁, 𝑋₂, and 𝑋₃𝜕𝑋=𝜕𝜕𝑡+𝛼𝜕𝜕𝑥+𝛽𝜕𝑦 (25) can be used to obtain a reduced conserved form. The generator, 𝑋, has a canonical form 𝑋=𝜕/𝜕𝑞 when 𝑑𝑡1=𝑑𝑥𝛼=𝑑𝑦𝛽=𝑑𝑢0=𝑑𝑟0=𝑑𝑠0=𝑑𝑞1=𝑑𝑣0, (26) and thus the canonical variables are 𝑣().𝑟=𝑦−𝛽𝑡,𝑠=𝑥−𝛼𝑡,𝑞=𝑡,𝑟,𝑠=𝑢𝑡,𝑥,𝑦 (27) The formula (10) for the reduced conserved form in terms of variables (𝑡,𝑥,𝑦) and (𝑟,𝑠,𝑞) can be expressed as 𝑇^𝑟₃𝑇^𝑠₃𝑇^𝑞₃=𝐽𝐴^−1𝑇𝑇^𝑡₃𝑇^𝑥₃𝑇^𝑦₃, (28) where 𝐴⁻¹ from (11) is given by 𝐴⁻¹=𝐷_𝑡𝑟𝐷_𝑡𝑠𝐷_𝑡𝑞𝐷_𝑥𝑟𝐷_𝑥𝑠𝐷_𝑥𝑞𝐷_𝑦𝑟𝐷_𝑦𝑠𝐷_𝑦𝑞,𝐽=det(𝐴). (29) Equations (28) and (29) for the conserved vector (24) results in 𝑇^𝑟₃=𝛽𝑣−𝑑𝑣_𝑠𝑟,𝑇^𝑠₃𝑣=𝛼𝑣−𝑎𝑣−𝑏²2−𝑐𝑣_𝑠𝑠,𝑇^𝑞₃=−𝑣, (30) and reduced conserved form is 𝐷_𝑟𝑇^𝑟₃+𝐷_𝑠𝑇^𝑠₃=0. (31) The generalized double reduction theorem reduced the third-order ZK equation (14) from the third-order PDE in terms of three independent variables (𝑡,𝑥,𝑦) to a system of two second-order PDES with two independent variables (𝑟,𝑠). 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: 𝑋₁=𝜕,𝑋𝜕𝑟²=𝜕.𝜕𝑠 (32) Since 𝑋₁ and 𝑋₂  satisfy the symmetry conservation law relation 𝑋𝑇^𝑟𝑇^𝑠−𝐷_𝑟𝜉^𝑟𝐷_𝑠𝜉^𝑟𝐷_𝑟𝜉^𝑠𝐷_𝑠𝜉^𝑠𝑇^𝑟𝑇^𝑠+𝐷_𝑟𝜉^𝑟+𝐷_𝑠𝜉^𝑠𝑇^𝑟𝑇^𝑠=0, (33) therefore 𝑋₁, and 𝑋₂ are the associated symmetries, and it is possible to find second reduction. A reduced conserved form can be obtained using 𝜕𝑌=𝜕𝜕𝑟+𝛾.𝜕𝑠 (34) The canonical form of generator 𝑌 is 𝑌=𝜕/𝜕𝑚 with the similarity variables 𝑛=𝛾𝑟−𝑠,𝑚=𝑟,𝑤(𝑛)=𝑣(𝑟,𝑠). (35) In this case, the formula (10) for the reduced conserved form results in 𝑇^𝑛₃𝑇^𝑚₃=𝐽𝐴^−1𝑇𝑇^𝑟₃𝑇^𝑠₃, (36) with 𝐴⁻¹=𝐷_𝑟𝑛𝐷_𝑟𝑚𝐷_𝑠𝑛𝐷_𝑠𝑚. (37) Equations (36) and (37) for the conserved vector (30) take the following form: 𝑇^𝑛₃𝑇^𝑚₃=𝛾−110𝛽𝑣−𝑑𝑣_s𝑟𝑣𝛼𝑣−𝑎𝑣−𝑏²2−𝑐𝑣_𝑠𝑠. (38) Equation (38) expressed in terms of variable 𝑛 becomes 𝑇^𝑛₃=𝑏𝛾𝛽−𝛼+𝑎𝑤+2𝑤²+𝛾²𝑑+𝑐𝑤_𝑛𝑛,𝑇^𝑚₃=𝛽𝑤+𝛾𝑑𝑤_𝑛𝑛, (39) and the reduced conserved form is 𝐷_𝑛𝑇^𝑛₃=0. (40) Equation (40) gives 𝑇^𝑛₃=𝑘₁, and (39) can be written as 𝑏𝛾𝛽−𝛼+𝑎𝑤+2𝑤²+𝛾²𝑑+𝑐𝑤_𝑛𝑛=𝑘₁. (41) The symmetries 𝑋₁, 𝑋₂, and 𝑋₃ are associated with the conserved vector 𝑇₃ only when 𝑓(𝑦)=1. For this case, the generalized double reduction theorem is applied twice to ZK equation (14) and it is reduced to second-order 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 is√±3𝛾²𝑑+𝑐𝛾²𝑑+𝑐3𝛼𝑤²−3𝑎𝑤²−𝑏𝑤³−3𝛾𝛽𝑤²+3𝛾²𝑑𝑐₁+3𝑐𝑐₁+6𝑘𝑤𝑑𝑤−𝑛−𝑐₂=0. (42)Now, we compute the explicit solutions of (41) by utilizing the Sine-Cosine method [18–20]. The solution of (41) can be expressed in the form 𝑤(𝑛)=𝜈cos^𝜅(𝜔𝑛), (43) or 𝑤(𝑛)=𝜈sin^𝜅(𝜔𝑛), (44) where 𝜈, 𝜅≠0 and 𝜔 are parameters need to be determined.

Substituting the values of 𝑤 from (43) and setting 𝑘₁=0 in (41) yields 𝐴𝜈cos^𝜅()+𝑏𝜔𝑛2𝜈²cos^2𝜅(𝜔𝑛)+𝐵𝜈𝜅²𝜔²cos^𝜅−2(𝜔𝑛)−𝐵𝜈𝜅²𝜔²cos^𝜅(𝜔𝑛)−𝐵𝜈𝜅𝜔²cos^𝜅−2(𝜔𝑛)=0, (45) where 𝐴=𝛾𝛽−𝛼+𝑎 and 𝐵=𝛾²𝑑+𝑐. Equation (45) is satisfied if 𝜅−2=2𝜅,𝛾𝛽−𝛼+𝑎−4𝜇²𝛾²𝑑−4𝜇²1𝑐=0,2𝑏𝜆+6𝜇²𝛾²𝑑+6𝜇²𝑐=0. (46) Ultimately, the solution of algebraic system (46) yields the solution of (41) and is given by 𝑤(𝑛)=−3𝛾𝛽+𝑎−𝛼𝑏sec²𝛾𝛽+𝑎−𝛼4𝑐+4𝑑𝛾²𝑛,𝑛=𝛾𝑦−𝑥+𝑡𝛼−𝛽𝛾,𝑤=𝑢. (47) Similarly, using the Sine function (44) one can easily obtain the solution of (41) as 𝑤(𝑛)=−3𝛾𝛽+𝑎−𝛼𝑏csc²𝛾𝛽+𝑎−𝛼4𝑐+4𝑑𝛾²𝑛, (48) where 𝑛=𝛾𝑦−𝑥+𝑡(𝛼−𝛾𝛽), 𝑤=𝑢. The solutions (47) and (48) can be finally expressed in terms of original variables as 𝑢𝑡,𝑥,𝑦=−3𝛾𝛽+𝑎−𝛼𝑏×sec²𝛾𝛽+𝑎−𝛼4𝑐+4𝑑𝛾²,𝑢𝛾𝑦−𝑥+𝑡𝛼−𝛾𝛽𝑡,𝑥,𝑦=−3𝛾𝛽+𝑎−𝛼𝑏×csc²𝛾𝛽+𝑎−𝛼4𝑐+4𝑑𝛾²𝛾𝑦−𝑥+𝑡𝛼−𝛾𝛽, (49)(50) and these constitute the exact solutions of ZK equation (14).

3.2. Reduction via 𝑇₂ Using Combination of Symmetries 𝑋₁, 𝑋₂, and 𝑋₃

The symmetries 𝑋₁,𝑋₂, and𝑋₃ are associated with the conserved vector 𝑇₂, and a reduced conserved form can be obtained by the combination 𝑋 given in (25). For the conserved vector 𝑇₂ in terms of canonical variables (27), we have 𝑇^𝑟₂𝑣=𝛽²2−𝑑𝑣𝑣_𝑟𝑠2+𝑑𝑣_𝑟𝑣_𝑠2,𝑇^𝑠₂𝑣=𝛼²2−𝑏𝑣³3−𝑎𝑣²2−𝑐𝑣𝑣_𝑠𝑠+𝑐𝑣²_𝑠2−𝑑𝑣𝑣_𝑟𝑟2,𝑇^𝑞₂𝑣=−²2, (51) and reduced conserved form is 𝐷_𝑟𝑇^𝑟₂+𝐷_𝑠𝑇^𝑠₂=0. The reduced conserved form admits the symmetries (32) and these symmetries satisfy the symmetry conservation law relation (33) for the conserved vectors 𝑇^𝑟₂ and 𝑇^𝑠₂. The combination of symmetries yields the generator 𝑌 and the similarity variables (35). Equations (36) and (37) for the conserved vector (51) gives 𝑇^𝑛₂=𝛽𝛾+𝑎−𝛼2𝑤²+𝛾²𝑑+𝑐𝑤𝑤_𝑛𝑛+𝑏𝑤³3−𝛾²𝑑+𝑐2𝑤²_𝑛,𝑇^𝑚₂𝑤=−𝛽²2−𝑑𝛾𝑤𝑤_𝑛𝑛2+𝑑𝛾𝑤²_𝑛2. (52) The reduced conserved form is 𝐷_𝑛𝑇^𝑛₂=0, (53) and this yields 𝛽𝛾+𝑎−𝛼2𝑤²+𝛾²𝑑+𝑐𝑤𝑤_𝑛𝑛+𝑏𝑤³3−𝛾²𝑑+𝑐2𝑤²_𝑛=𝑘₂, (54) where 𝑘₂ is a constant. Using Maple, the solution of (54) is√±3𝑑𝛾²+𝑐𝑑𝛾²+𝑐3𝛼𝑤²−3𝑎𝑤²−3𝛾𝛽𝑤²−𝑏𝑤³+3𝑐𝑐₃𝑤+3𝑐₃𝑑𝛾²𝑤−6𝑘₂𝑑𝑤−𝑛−𝑐₄=0,𝑛=𝛾𝑦−𝑥+𝑡𝛼−𝛾𝛽,𝑤=𝑢, (55)whereas (54) gives the same solution by employing Sine-Cosine method as we have obtained in the previous case. The symmetries 𝑋₁, 𝑋₂, and 𝑋₃ are associated with the conserved vector 𝑇₂, 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 𝑇₃ Using Symmetry 𝑋₅

The generator, 𝑋₅, has canonical form 𝑋=𝜕/𝜕𝑞 if 𝑑𝑡=3𝑡𝑑𝑥=𝑥+2𝑎𝑡𝑑𝑦𝑦=𝑑𝑢=−2𝑢𝑑𝑟0=𝑑𝑠0=𝑑𝑞1=𝑑𝑣0, (56) and thus we have 1𝑞=3𝑦ln𝑡,𝑟=𝑡^1/3(),𝑠=𝑥−𝑎𝑡𝑡^1/3,𝑣(𝑟,𝑠)=𝑢𝑡,𝑥,𝑦𝑡^2/3. (57) Equations (28) and (29) for the conserved vector 𝑇₃ in terms of canonical variables (57) result in 𝑇^𝑟₃=𝑟𝑣−3𝑑𝑣_𝑟𝑠,𝑇^𝑠₃3=𝑠𝑣−2𝑏𝑣²−3𝑐𝑣_𝑠𝑠, (58) and reduced conserved form is 𝐷_𝑟𝑇^𝑟₃+𝐷_𝑠𝑇^𝑠₃=0. 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 Sine-Cosine 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 Exp-function 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 𝑢_𝑡𝑥+6𝑢𝑢_𝑥𝑥+6𝑢²_𝑥+6𝑢²𝑢_𝑥𝑥+12𝑢𝑢²_𝑥+𝑢_{𝑥𝑥𝑥𝑥}+𝑢_𝑦𝑦=0. (59) The Lie symmetry generator determining equation for Gardner KP equation (59) is 𝑋^[4]𝑢_𝑡𝑥+6𝑢𝑢_𝑥𝑥+6𝑢²_𝑥+6𝑢²𝑢_𝑥𝑥+12𝑢𝑢²_𝑥+𝑢_{𝑥𝑥𝑥𝑥}+𝑢_𝑦𝑦(59)=0, (60) where 𝑋^[4] is the fourth prolongation. Solving (60), after expansion, we have 𝑋₁=𝜕𝜕𝑡,𝑋₂=𝜕𝜕𝑥,𝑋₃=𝜕,𝑋𝜕𝑦₄𝜕=𝑦𝜕𝜕𝑥−2𝑡,𝑋𝜕𝑦₅𝜕=−6𝑡+()𝜕𝜕𝑡−2𝑥+6𝑡𝜕𝜕𝑥−4𝑦+()𝜕𝜕𝑦2𝑢+1,𝜕𝑢 (61) as the Lie symmetry generators for the Gardner KP equation (59). Consider the multipliers of the form Λ=Λ(𝑡,𝑥,𝑢) for (59). The determining equation for the multipliers is 𝛿Λ𝑢𝛿𝑢^𝑡𝑥+6𝑢𝑢_𝑥𝑥+6𝑢²_𝑥+6𝑢²𝑢_𝑥𝑥+12𝑢𝑢²_𝑥+𝑢_{𝑥𝑥𝑥𝑥}+𝑢_𝑦𝑦=0. (62) Equation (62) finally presents Λ_𝑢=0,Λ_𝑥𝑥=0,Λ_𝑥𝑦𝑦Λ=0,_{𝑦𝑦𝑦𝑦}=0,Λ_𝑡𝑥+Λ_𝑦𝑦=0, (63) and this results in four multipliers. The multipliers and conserved vectors are presented in Table 2.

Table 2 

Multipliers and conserved vectors for (59).

The symmetry conservation law relation (23) is not satisfied for the conserved vectors 𝑇₁, 𝑇₂, and 𝑇₃. The symmetries 𝑋₁, 𝑋₂, and 𝑋₃ are associated with the conserved vector 𝑇₄ if 𝑓(𝑡)=1. Thus we can get a reduced conserved form by the combination of 𝑋 given in (25). Equations (28) and (29) for the conserved vector 𝑇₄ in terms of canonical variables (27) yield the following three components of 𝑇₄: 𝑇^𝑟₄=𝛽𝑣_𝑠−𝑣_𝑟,𝑇^𝑠₄=𝛼𝑣_𝑠−𝑣_𝑠𝑠𝑠−6𝑣²𝑣_𝑠−6𝑣𝑣_𝑠,𝑇^𝑞₄=−𝑣_𝑠, (64) and reduced conserved form is 𝐷_𝑟𝑇^𝑟₄+𝐷_𝑠𝑇^𝑠₄=0. (65) The reduced conservation law admits the following symmetries: 𝑋₁=𝜕,𝑋𝜕𝑟²=𝜕.𝜕𝑠 (66) The symmetries 𝑋₁ and 𝑋₂ satisfy the symmetry conservation law relation (33) for the conserved vectors 𝑇^𝑟₄ and 𝑇^𝑠₄. Taking the combination of these symmetries yields the same generator 𝑌 as given in (34), and the canonical form 𝑌=𝜕/𝜕𝑚 can be obtained from similarity variables (35). Using formula (36), one has the following two components of 𝑇₄: 𝑇^𝑛₄=𝛼−𝛽𝛾−𝛾²𝑤_𝑛−6𝑤²𝑤_𝑛−6𝑤𝑤_𝑛−𝑤_𝑛𝑛𝑛,𝑇^𝑚₄=−𝛽+𝛾𝑤_𝑛. (67) The reduced conserved form satisfies 𝐷_𝑛𝑇^𝑛=0, and we have 𝛼−𝛽𝛾−𝛾²𝑤_𝑛−6𝑤²𝑤_𝑛−6𝑤𝑤_𝑛−𝑤_𝑛𝑛𝑛=𝑘₃. (68) The integration of (68) provides 𝛼−𝛽𝛾−𝛾²𝑤−2𝑤³−3𝑤²−𝑤_𝑛𝑛=𝑘₃𝑛+𝑘₄. (69) 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 𝑘₃=0: ±1𝛼−𝛽𝛾−𝛾²𝑤²−2𝑤³−𝑤⁴−2𝑘₄𝑤+𝑐₅𝑑𝑤−𝑛−𝑐₆=0, (70) where 𝑛=(𝛼−𝛽𝛾)𝑡+𝛾𝑦−𝑥, 𝑤=𝑢 and 𝑐₅, 𝑐₆ and 𝑘₄ are constants.

For explicit solution, we apply the first integral method to the reduced form (69). We substitute 𝑤=𝑋 and 𝑤^′=𝑌 with 𝑘₃=𝑘₄=0 which converts (69) into the following system of ODEs: 𝑋^′𝑌=𝑌,_′=𝐴𝑋−3𝑋²−2𝑋³, (71) where 𝐴=𝛼−𝛽𝛾−𝛾². Next, we apply the division theorem to seek the first integral to (71). Assume that 𝑋=𝑋(𝑛) and 𝑌=𝑌(𝑛) are the nontrivial solutions to (71) and 𝑝(𝑋(𝑛),𝑌(𝑛))=𝑎_𝑖(𝑋(𝑛))𝑌^𝑖=0,𝑖=1,2,…,𝑚 (72) is an irreducible polynomial in 𝑐|𝑋,𝑌| such that 𝑝(𝑋(𝑛),𝑌(𝑛))=𝑎_𝑖(𝑋(𝑛))𝑌(𝑛)^𝑖=0,𝑖=1,2,…,𝑚, (73) where 𝑎_𝑖(𝑋), (𝑖=1,2,…,𝑚) are polynomials of 𝑋 and all relatively prime in 𝑐|𝑋,𝑌|, 𝑎_𝑚(𝑋)≠0. Equation (73) is also called the first integral to (71). Suppose that 𝑚=1 in (73). By division theorem, there exist polynomials 𝐻(𝑋,𝑌)=𝑔(𝑋)+ℎ(𝑋)𝑌 in 𝑐|𝑋,𝑌| such that 𝑑𝑝=𝑑𝑛𝜕𝑝𝜕𝑋𝜕𝑋+𝜕𝑛𝜕𝑝𝜕𝑌𝜕𝑌=𝑔(𝑋)(𝑋)𝑌𝑎𝜕𝑛+ℎ₀(𝑋)+𝑎₁(𝑋)𝑌, (74) or 𝑎^′₀(𝑋)+𝑎^′₁(𝑋)(𝑌)𝑌+𝑎₁(𝑋)𝑌^′=𝑔(𝑋)+ℎ(𝑋)𝑌𝑎₀(𝑋)+𝑎₁(𝑋)𝑌. (75) Substituting 𝑌^′ from (71) in (75) and then separating with respect to powers of 𝑌, we obtain 𝑎^′₁(𝑋)=ℎ(𝑋)𝑎₁(𝑋),𝑎^′₀(𝑋)=𝑔(𝑋)𝑎₁(𝑋)+ℎ(𝑋)𝑎₀(𝑋),𝑎₁(𝑋)𝐴𝑋−3𝑋²−2𝑋³=𝑔(𝑋)𝑎₀(𝑋). (76) Solving the system (76) for 𝑎₀ and 𝑎₁ and then substituting it into (73) yields an ODE which finally gives 𝑢=1𝑡,𝑥,𝑦𝑐₅𝑒^{𝜄(𝛾𝑦−𝛾𝛽𝑡−𝑥+(𝛾𝛽+𝛾2−1)𝑡)},𝑢=1−1𝑡,𝑥,𝑦𝑐₆𝑒^{−𝜄(𝛾𝑦−𝛾𝛽𝑡−𝑥+(𝛾𝛽+𝛾2−1)𝑡)}.−1 (77) 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 third-order ODE (68). On integration, the third order ODE (68) is further reduced to second-order 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 second-order ODE (69). The exact solutions derived here are different from class of multiple-soliton 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 𝑢_𝑡𝑥+𝑎𝑢𝑢_𝑥𝑥+𝑎𝑢²_𝑥+2𝑑𝑢𝑢²_𝑥+𝑑𝑢²𝑢_𝑥𝑥+𝑏𝑢_{𝑥𝑥𝑥𝑥}+𝑐𝑢_𝑥𝑥+𝑢_𝑦𝑦=0, (78) where 𝑎, 𝑏, 𝑐, and 𝑑 are plasma parameters. For (78) the multipliers of the form Λ=Λ(𝑡,𝑥,𝑦,𝑢) are considered. The multipliers and conserved vectors are given in Table 3. The MKP equation (78) has the following Lie symmetry generators: 𝑋₁=𝜕𝜕𝑡,𝑋₂=𝜕,𝑋𝜕𝑥₃=𝜕𝜕𝑦,𝑋₄𝜕=𝑦𝜕𝜕𝑥−2𝑐𝑡,𝑋𝜕𝑦₅𝜕=−6𝑑𝑡+𝜕𝑡−2𝑑𝑥−4𝑡𝑐𝑑+𝑎²𝑡𝜕𝜕𝜕𝑥−4𝑑𝑦+()𝜕𝜕𝑦2𝑑𝑢+𝑎.𝜕𝑢 (79) Only the Lie symmetries 𝑋₁, 𝑋₂, and 𝑋₃ are associated with the conserved vector 𝑇₄ when 𝑓(𝑡)=1. By using their combination as we have done in Section 3, and with the aid of 𝑇₄, we obtain −𝛽𝛾−𝑐𝛾²𝑎+𝛼−𝑐𝑤−2𝑤²−𝑏𝑤_𝑛𝑛−𝑑3𝑤³=𝑘₅𝑛+𝑘₆. (80) A particular solution of (80) can be found for the case 𝑘₅=0 and is given by ±6𝑏−6𝑏6𝛽𝛾+𝑐𝛾²−𝛼+𝑐𝑤²+2𝑎𝑤³+𝑑𝑤⁴−12𝑘₆𝑤−6𝑏𝑐₅𝑑𝑤−𝑛−𝑐₆=0, (81)where 𝑛=𝛾𝑦−𝛾𝛽𝑡−𝑥+𝛼𝑡, 𝑤=𝑢.

Table 3 

Multipliers and conserved vectors for (78).

A class of solitary wave solutions were reported in [17] using (80), whereas the above solution is not reported there.

6. Conclusions

The generalized double reduction theorem provides a powerful tool in constructing reduced forms and exact solutions. It enables a systematic way to find not only the transformations providing traveling wave solutions but also other types of transformations. These transformations reduce a nonlinear system of 𝑞th-order PDEs with 𝑛 independent and 𝑚 dependent variables to a nonlinear system of (𝑞−1)th-order ODEs provided that in every reduction at least one symmetry is associated with a nontrivial conserved vector. The reduced ODE can be solved either analytically or numerically to derive exact or approximate solutions. It is interesting that the transformations yielding traveling wave solutions can give sometimes more than one reduced form, and one can use the simple one to find exact solution.

The Lie symmetries, conservation laws, reduced forms and new exact solutions of (2+1) dimensional ZK, GKP, and MKP equations were derived. First of all ZK equation was considered, and the Lie symmetries and conservation laws were constructed. Multiplier approach yielded three conserved vectors. The symmetry conservation laws relationship was used to determine symmetries associated with the conserved vectors. Three symmetries were associated with the conserved vector 𝑇₃ if 𝑓(𝑦)=1. The generalized double reduction theorem was applied twice to ZK equation to convert it to a second-order ordinary differential equation (41). Thus third-order (2+1) dimensional ZK equation was reduced to a second-order ordinary differential equation in terms of canonical variables. Furthermore, one implicit solution was found for (41) which constituted the exact solution of ZK equation. The Sine-Cosine method was applied to the reduced second-order ODE (41), and two explicit solutions were computed for ZK equation. Likewise, symmetries 𝑋₁, 𝑋₂, and 𝑋₃ were associated with the conserved vector 𝑇₂ and reduced conserved form of ZK equation was derived. One implicit solution was constructed using Maple, whereas same explicit solutions were obtained as in the previous case. The symmetry 𝑋₅ was associated with the conserved vector 𝑇₃, and ZK equation was reduced to second-order system (58). It was not possible to further reduce system (58) because it does not admit any symmetry associated with it, however one can apply approximate methods or numerical techniques to compute the approximate solutions.

The Lie symmetries and conservation laws for GKP equation were established. The GKP equation was reduced to a third-order ODE (68), and on integration it was further reduced to a second-order ODE (69). An implicit solution for (70) was found for GKP equation. Two explicit solutions of GKP equation were derived utilizing the first integral method. For MKP equation, we derived the Lie symmetries, conservation laws, reduced form, and one implicit solution.

The solutions found here are new and not found in literature. Due to the lack of experimental basis, the derived solutions cannot be interpreted physically but in applied mathematics these will play a vital role for numerical simulations.