Some Reduction and Exact Solutions of a Higher-Dimensional Equation

Abstract

The conservation laws of the (3+1)-dimensional Zakharov-Kuznetsov equation were obtained using Noether’s theorem after an interesting substitution 𝑢=𝑣_𝑥 to the equation. Then, with the aid of an obtained conservation law, the generalized double reduction theorem was applied to this equation. It can be verified that the reduced equation is a second order nonlinear ODE. Finally, some exact solutions of the Zakharov-Kuznetsov equation were constructed after solving the reduced equation.

1. Introduction

As we know that conservation laws play an important role in the study of nonlinear partial differential equations (NLPDEs) [1–3], especially in the reduction and solution of a NLPDE, an elegant and constructive way to derive conservation laws is Noether’s approach [4, 5]. The application of Noether’s approach depends on a suitable Lagrangian. Yet there also exists some PDEs which do not have a Lagrangian, for example, the scalar evolution differential equations. Also there have several methods which do not relay on the knowledge of a Lagrangian. The most recent approach, due to Kara and Mahomed, is partial Noether’s approach [6]. It works like the Noether’s approach for PDEs with or without a Lagrangian. Different approaches to construct conservation laws were discussed in detail [7].

The fundamental relationship between symmetries and conservation laws of a PDE was given out by Kara and Mahomed in [8]. For the case of a PDE with two independent variables, using the definition of the association of symmetries with conservation laws, Sjöberg developed a theorem called double reduction method [9] which leads from a 𝑞th order PDE to a 𝑞−1th order ordinary differential equation (ODE). Recently, this method has been generalized in [10] and applied to a (1+2)-dimensional wave equation [11].

In this paper, we focus on an equation presented in [12] which shows that 𝑢_𝑡+𝑎𝑢𝑢_𝑥+𝑏𝑢_𝑥𝑥𝑥+𝑢_𝑥𝑦𝑦+𝑢_𝑥𝑧𝑧=0. (1)

In [12], the authors investigated an isothermal multicomponent magnetized plasma and firstly derived this equation. In a very recent paper [13], the authors firstly studied the symmetry group of this equation. The one-, two-, and three-parameter optimal systems of group-invariant solutions were also given out. Then, based on the obtained optimal system, they derived the reductions and some new solutions of this equation.

This paper is arranged as follows: in Section 2, we first briefly present some notation and pertinent results which will be used in this paper. In Section 3, the conservation laws of this equation will be constructed by using Noether’s approach after an interesting substituting. Then in Section 4, with the aid of an obtained conservation law and its associated symmetries, the generalized double reduction method will be applied to the Zakharov-Kuznetsov equation. And in Section 5, some exact solutions will be constructed after solving the corresponding reduced nonlinear ODE. Finally, some conclusions and discussions are given in Section 6.

2. Notation and Preliminaries

We briefly present the notation and pertinent results which we utilize below. In this section, the summation convention is used whenever appropriate.

2.1. Fundamental Operators and Their Relationship

Consider that a 𝑘-order system of PDEs with 𝑛 independent variables 𝑥=(𝑥¹,𝑥²,…,𝑥^𝑛) and 𝑚 dependent variables 𝑢=(𝑢¹,𝑢²,…,𝑢^𝑚) reads 𝐸_𝛼𝑥,𝑢,𝑢₍₁₎,…,𝑢_(𝑘)=0,𝛼=1,2,…,𝑚 (2) which is assumed to be of maximal rank and locally solvable.

The collection of 𝑟th-order derivatives is denoted by 𝑢_(𝑟), and the derivatives of 𝑢^𝛼 with respect to 𝑥^𝑖 are 𝑢^𝛼_𝑖=𝐷_𝑖(𝑢^𝛼), 𝑢^𝛼_𝑖𝑗=𝐷_𝑗𝐷_𝑖(𝑢^𝛼), where 𝐷_𝑖=𝜕𝜕𝑥^𝑖+𝑢^𝛼_𝑖𝜕𝜕𝑢^𝛼+𝑢^𝛼_𝑖𝑗𝜕𝜕𝑢^𝛼_𝑗+⋯,𝑖=1,2,…,𝑛 (3) is the total derivative operator with respect to 𝑥^𝑖.

The following results are well known which can be found in many literatures.

The Euler-Lagrange operator is defined as 𝛿𝛿𝑢^𝛼=𝜕𝜕𝑢^𝛼+^∞_𝑗=1(−1)^𝑗𝐷_𝑖1⋯𝐷_𝑖𝑗𝜕𝜕𝑢^𝛼_{𝑖1⋯𝑖𝑗},𝛼=1,2,…,𝑚 (4) and the Lie-Bäcklund operator is given by 𝑋=𝜉^𝑖𝜕𝜕𝑥^𝑖+𝜂^𝛼𝜕𝜕𝑢^𝛼+^∞_𝑗=1𝜁^𝛼_{𝑖1𝑖2⋯𝑖𝑗}𝜕𝜕𝑢^𝛼_{𝑖1⋯𝑖𝑗}, (5) where 𝜁^𝛼_{𝑖1𝑖2⋯𝑖𝑗} are determined by the following formulation: 𝜁^𝛼_𝑖=𝐷_𝑖𝜂^𝛼−𝑢^𝛼_𝑠𝐷_𝑖𝜉^𝑠,𝜁^𝛼_{𝑖1𝑖2⋯𝑖𝑗}=𝐷_𝑖𝑗𝜁^𝛼_{𝑖1𝑖2⋯𝑖𝑗−1}−𝑢^𝛼_{𝑠𝑖1𝑖2⋯𝑖𝑗−1}𝐷_𝑖𝑗𝜉^𝑠,𝑗>1. (6) And a Lie-Bäcklund operator can also be written in characteristic form as 𝑋=𝜉^𝑖𝐷_𝑖+𝑊^𝛼𝜕𝜕𝑢^𝛼+^∞_𝑗=1𝐷_𝑖1⋯𝐷_𝑖𝑗𝑊^𝛼𝜕𝜕𝑢^𝛼_{𝑖1⋯𝑖𝑗}, (7) where 𝑊^𝛼=𝜂^𝛼−𝜉^𝑗𝑢^𝛼_𝑗, 𝛼=1,2,…,𝑚. are the Lie characteristic functions.

A Noether operator associated with a Lie-Bäcklund operator 𝑋 is defined as 𝑁^𝑖=𝜉^𝑖+𝑊^𝛼𝛿𝛿𝑢^𝛼_𝑖+^∞_𝑗=1𝐷_𝑖1⋯𝐷_𝑖𝑗𝑊^𝛼𝛿𝛿𝑢^𝛼_{𝑖𝑖1⋯𝑖𝑗},𝑖=1,2,…,𝑛 (8) in which, the Euler-Lagrange operators with respect to derivatives of 𝑢^𝛼 can be achieved by replacing 𝑢^𝛼 with the corresponding derivatives. For example, the Euler-Lagrange operator 𝛿/𝛿𝑢^𝛼_𝑖 is given by 𝛿𝛿𝑢^𝛼_𝑖=𝜕𝜕𝑢^𝛼_𝑖+^∞_𝑗=1(−1)^𝑗𝐷_𝑖1⋯𝐷_𝑖𝑗𝜕𝜕𝑢^𝛼_{𝑖𝑖1⋯𝑖𝑗},𝑖=1,2,…,𝑛,𝛼=1,2,…,𝑚, (9) and the other Euler-Lagrange operators with respect to higher order derivatives can be derived in the similar way.

2.2. Conservation Law

An 𝑛-tuple 𝑇=(𝑇¹,𝑇²,…,𝑇^𝑛), 𝑇^𝑖∈𝜗,  𝑖=1,2,…,𝑛, is a conserved vector of the system (2) if 𝑇^𝑖 satisfies 𝐷_𝑖𝑇^𝑖₍₂₎=0, (10) where 𝜗 is the space of differential functions.

2.3. Noether’s Theorem

If there exists a function 𝐿=𝐿(𝑥,𝑢,𝑢₍₁₎,…,𝑢_(𝑙)), 𝑙≤𝑘, such that (2) is equivalent to 𝛿𝐿𝛿𝑢^𝛼=0,𝛼=1,2,…,𝑚 (11) then 𝐿 is called a Lagrangian of (2) and (11) is the corresponding Euler-Lagrange system.

If a Lie-Bäcklund operator 𝑋 defined in (5) satisfies 𝑋(𝐿)+𝐿𝐷_𝑖𝜉^𝑖=𝐷_𝑖𝐵^𝑖 (12) for some vectors 𝐵=(𝐵¹,𝐵²,…,𝐵^𝑛), 𝐵^𝑖(𝑥¹,…,𝑥^𝑛,𝑢¹,…,𝑢^𝑚)∈𝜗, 𝑖=1,2,…,𝑛, then it is a Noether symmetry generator associated with the corresponding Lagrangian 𝐿.

Having determined the Noether symmetry generators, conservation laws of the Euler-Lagrange system can be constructed through the following theorem in an elegant way.

Theorem 1. If a Lie-Bäcklund operator 𝑋 defined in (5) is a Noether symmetry generator associated with a Lagrangian 𝐿 of an Euler-Lagrange system, then there corresponds a vector 𝑇=(𝑇¹,𝑇²,…,𝑇^𝑛) with 𝑇^𝑖 given by 𝑇^𝑖=𝐵^𝑖−𝑁^𝑖𝐿=𝐵^𝑖−𝜉^𝑖𝐿−𝑊^𝛼𝛿𝐿𝛿𝑢^𝛼_𝑖−^∞_𝑗=1𝐷_𝑖1⋯𝐷_𝑖𝑗𝑊^𝛼𝛿𝐿𝛿𝑢^𝛼_{𝑖𝑖1⋯𝑖𝑗},𝑖=1,2,…,𝑛 (13) which is a conserved vector of the Euler-Lagrange system, where the characteristics 𝑊=(𝑊¹,𝑊²,…,𝑊^𝑚),  𝑊^𝑖∈𝜗 of the Noether symmetry generator are also the characteristics of the obtained conservation law.

2.4. Generalized Double Reduction Theorem

Definition 2 (see [8]). A Lie-Bäcklund symmetry generator 𝑋 defined in (5) is said to be associated with a conserved vector 𝑇 of the system (2) if 𝑋 and 𝑇 satisfy the following identity: 𝑋𝑇^𝑖+𝑇^𝑖𝐷_𝑘𝜉^𝑘−𝑇^𝑘𝐷_𝑘𝜉^𝑖=0,𝑖=1,…,𝑛. (14)

Theorem 3 (see [14]). Suppose that 𝑋 is any Lie-Bäcklund symmetry generator of the system (2) and 𝑇^𝑖, 𝑖=1,…,𝑛 are the components of conserved vector of (2). Then 𝑇^∗𝑖=𝑇^𝑖,𝑋=𝑋𝑇^𝑖+𝑇^𝑖𝐷_𝑗𝜉^𝑗−𝑇^𝑗𝐷_𝑗𝜉^𝑖,𝑖=1,…,𝑛 (15) also constitute the components of a conserved vector for (2); that is, 𝐷_𝑖𝑇^∗𝑖₍₂₎=0. (16)

Theorem 4 (see [10]). Suppose that 𝐷_𝑖𝑇^𝑖=0 is a conservation law of the system (2). Then, under a contact transformation, there exists function 𝑇^𝑖 such that 𝐽𝐷_𝑖𝑇^𝑖=𝐷_𝑖𝑇^𝑖, where 𝑇^𝑖 is given by the following formula: 𝑇¹𝑇²⋮𝑇^𝑛=𝐽𝐴^−1𝑇𝑇¹𝑇²⋮𝑇^𝑛,𝐽𝑇¹𝑇²⋮𝑇^𝑛=𝐴^𝑇𝑇¹𝑇²⋮𝑇^𝑛 (17) in which 𝐷𝐴=¹𝑥₁𝐷₁𝑥₂⋯𝐷₁𝑥_𝑛𝐷₂𝑥₁𝐷₂𝑥₂⋯𝐷₂𝑥_𝑛𝐷⋮⋮⋮⋮_𝑛𝑥₁𝐷_𝑛𝑥₂⋯𝐷_𝑛𝑥_𝑛,𝐴⁻¹=𝐷₁𝑥₁𝐷₁𝑥₂⋯𝐷₁𝑥_𝑛𝐷₂𝑥₁𝐷₂𝑥₂⋯𝐷₂𝑥_𝑛𝐷⋮⋮⋮⋮_𝑛𝑥₁𝐷_𝑛𝑥₂⋯𝐷_𝑛𝑥_𝑛 (18) and 𝐽=det(𝐴).

Theorem 5 (fundamental theorem on generalized double reduction [10]). Suppose that 𝐷_𝑖𝑇^𝑖=0 is a conservation law of the system (2). Then, under a similarity transformation of a symmetry 𝑋 of the system (2), there exist functions 𝑇^𝑖 such that 𝑋 is still a symmetry for the PDE 𝐷_𝑖𝑇^𝑖=0 and 𝑋𝑇¹𝑋𝑇²⋮𝑋𝑇^𝑛=𝐽𝐴^−1𝑇𝑇¹𝑇,𝑋₂⋮𝑇,𝑋_𝑛,,𝑋 (19) in which 𝐷𝐴=¹𝑥₁𝐷₁𝑥₂⋯𝐷₁𝑥_𝑛𝐷₂𝑥₁𝐷₂𝑥₂⋯𝐷₂𝑥_𝑛𝐷⋮⋮⋮⋮_𝑛𝑥₁𝐷_𝑛𝑥₂⋯𝐷_𝑛𝑥_𝑛,𝐴⁻¹=𝐷₁𝑥₁𝐷₁𝑥₂⋯𝐷₁𝑥_𝑛𝐷₂𝑥₁𝐷₂𝑥₂⋯𝐷₂𝑥_𝑛𝐷⋮⋮⋮⋮_𝑛𝑥₁𝐷_𝑛𝑥₂⋯𝐷_𝑛𝑥_𝑛, (20) and 𝐽=det(𝐴).

Corollary 6 (necessary and sufficient condition for reduced conserved form [10]). The conserved form 𝐷_𝑖𝑇^𝑖=0 of the system (2) 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]|₍₂₎=0.

Corollary 7 (generalized double reduction theorem [10]). 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. Conservation Laws of the Zakharov-Kuznetsov Equation

Making a substitution 𝑢=𝑣_𝑥 to (1), we have 𝑣_𝑡𝑥+𝑎𝑣_𝑥𝑣_𝑥𝑥+𝑏𝑣_{𝑥𝑥𝑥𝑥}+𝑣_{𝑥𝑥𝑦𝑦}+𝑣_{𝑥𝑥𝑧𝑧}=0. (21) A Lagrangian for (21) satisfying the Euler-Lagrange equation 𝛿𝐿/𝛿𝑣=0 is given as 1𝐿=−2𝑣_𝑡𝑣_𝑥−𝑎6𝑣³_𝑥+𝑏2𝑣²_𝑥𝑥+12𝑣_𝑥𝑥𝑣_𝑦𝑦+12𝑣_𝑥𝑥𝑣_𝑧𝑧, (22) where the Euler-Lagrange operator 𝛿/𝛿𝑣 is given by 𝛿=𝜕𝛿𝑣𝜕𝑣−𝐷_𝑡𝜕𝜕𝑣_𝑡−𝐷_𝑥𝜕𝜕𝑣_𝑥−𝐷_𝑦𝜕𝜕𝑣_𝑦−𝐷_𝑧𝜕𝜕𝑣_𝑧+𝐷²_𝑥𝜕𝜕𝑣_𝑥𝑥+𝐷²_𝑦𝜕𝜕𝑣_𝑦𝑦+𝐷²_𝑧𝜕𝜕𝑣_𝑧𝑧+⋯. (23) The Lie-Bäcklund operator is (invoking (5) up to third order derivatives together with (6)) 𝑋=𝜉¹𝜕𝑡,𝑥,𝑦,𝑧,𝑣𝜕𝑡+𝜉²𝜕𝑡,𝑥,𝑦,𝑧,𝑣𝜕𝑥+𝜉³𝜕𝑡,𝑥,𝑦,𝑧,𝑣𝜕𝑦+𝜉⁴𝜕𝑡,𝑥,𝑦,𝑧,𝑣𝜕𝜕𝑧+𝜂𝑡,𝑥,𝑦,𝑧,𝑣𝜕𝑣+𝜂_𝑡⁽¹⁾𝜕𝜕𝑣_𝑡+𝜂_𝑥⁽¹⁾𝜕𝜕𝑣_𝑥+𝜂⁽²⁾_𝑥𝑥𝜕𝜕𝑣_𝑥𝑥+𝜂⁽²⁾_𝑦𝑦𝜕𝜕𝑣_𝑦𝑦+𝜂⁽²⁾_𝑧𝑧𝜕𝜕𝑣_𝑧𝑧, (24) in which 𝜂_𝑡⁽¹⁾=𝐷_𝑡(𝜂)−𝑣_𝑖𝐷_𝑡(𝜉^𝑖), 𝜂_𝑥⁽¹⁾=𝐷_𝑥(𝜂)−𝑣_𝑖𝐷_𝑥(𝜉^𝑖), 𝜂_𝑦⁽¹⁾=𝐷_𝑦(𝜂)−𝑣_𝑖𝐷_𝑦(𝜉^𝑖), 𝜂⁽²⁾_𝑥𝑥=𝐷_𝑥𝜂_𝑥⁽¹⁾−𝑣_𝑖𝑥𝐷_𝑥𝜉^𝑖,𝜂⁽²⁾_𝑦𝑦=𝐷_𝑦𝜂_𝑦⁽¹⁾−𝑣_𝑖𝑦𝐷_𝑦𝜉^𝑖,𝜂⁽²⁾_𝑧𝑧=𝐷_𝑧𝜂_𝑧⁽¹⁾−𝑣_𝑖𝑧𝐷_𝑧𝜉^𝑖. (25) The Lie-Bäcklund operator 𝑋 is a Noether symmetry generator associated with the Lagrangian 𝐿 (22) if there exists a vector 𝐵=(𝐵¹,𝐵²,𝐵³,𝐵⁴), such that 𝑋(𝐿)+𝐿𝐷_𝑡𝜉¹+𝐷_𝑥𝜉²+𝐷_𝑦𝜉³+𝐷_𝑧𝜉⁴=𝐷_𝑡𝐵¹+𝐷_𝑥𝐵²+𝐷_𝑦𝐵³+𝐷_𝑧𝐵⁴, (26) where 𝐵¹(𝑡,𝑥,𝑦,𝑧,𝑣), 𝐵²(𝑡,𝑥,𝑦,𝑧,𝑣), 𝐵³(𝑡,𝑥,𝑦,𝑧,𝑣), and 𝐵⁴(𝑡,𝑥,𝑦,𝑧,𝑣) are the gauge terms.

The expansion of (26) yields −12𝑣_𝑥𝜂_𝑡⁽¹⁾−12𝑣_𝑡+𝑎𝑣²_𝑥𝜂_𝑥⁽¹⁾+122𝑏𝑣_𝑥𝑥+𝑣_𝑦𝑦+𝑣_𝑧𝑧𝜂⁽²⁾_𝑥𝑥+12𝑣_𝑥𝑥𝜂⁽²⁾_𝑦𝑦+12𝑣_𝑥𝑥𝜂⁽²⁾_𝑧𝑧+−12𝑣_𝑡𝑣_𝑥−𝑎6𝑣³_𝑥+𝑏2𝑣²_𝑥𝑥+12𝑣_𝑥𝑥𝑣_𝑦𝑦+12𝑣_𝑥𝑥𝑣_𝑧𝑧×𝜉¹_𝑡+𝑣_𝑡𝜉¹_𝑣+𝜉²_𝑥+𝑣_𝑥𝜉²_𝑣+𝜉³_𝑦+𝑣_𝑦𝜉³_𝑣+𝜉⁴_𝑧+𝑣_𝑧𝜉⁴_𝑣=𝐵¹_𝑡+𝑣_𝑡𝐵¹_𝑣+𝐵²_𝑥+𝑣_𝑥𝐵²_𝑣+𝐵³_𝑦+𝑣_𝑦𝐵³_𝑣+𝐵⁴_𝑧+𝑣_𝑧𝐵⁴_𝑣. (27) Splitting (27) with respect to derivatives of 𝑣 results in an overdetermined system of equations for 𝜉¹,𝜉²,𝜉³,𝜉⁴,𝜂,𝐵¹,𝐵²,𝐵³, and 𝐵⁴. The solutions of this system yield the following Noether symmetries and gauge terms: 𝜉¹=3𝑐₁𝑡+𝑐₅,𝜉²=𝑐₁𝑥+𝑐₆𝑡+𝑐₇,𝜉³=𝑐₁𝑦+𝑐₂𝑧+𝑐₃,𝜉⁴=𝑐₁𝑧−𝑐₂𝑦+𝑐₄,𝜂=−𝑐₁1𝑣+𝑎𝑐₆,𝐵𝑥+𝛼𝑡,𝑧−𝑦𝑖+𝛽𝑡,𝑧+𝑦𝑖₁1=−𝑐2𝑎⁶𝑣+𝛾¹,𝐵𝑡,𝑥,𝑦,𝑧₂1=−2𝛼_𝑡+𝛽_𝑡𝑣+𝛾²,𝐵𝑡,𝑥,𝑦,𝑧₃=𝛾³𝑡,𝑥,𝑦,𝑧,𝐵⁴=𝛾⁴,𝛾𝑡,𝑥,𝑦,𝑧_1𝑡+𝛾²_𝑥+𝛾³_𝑦+𝛾⁴_𝑧=0, (28) where 𝑐_𝑖, 𝑖=1,2,…,7 are constants, 𝛼(𝑡,𝑧−𝑦𝑖) and 𝛽(𝑡,𝑧+𝑦𝑖) are two functions of their gauge terms. We can set 𝛾¹(𝑡,𝑥,𝑦,𝑧)=𝛾²(𝑡,𝑥,𝑦,𝑧)=𝛾³(𝑡,𝑥,𝑦,𝑧)=𝛾⁴(𝑡,𝑥,𝑦,𝑧)=0 as they contribute to the trivial part of the conserved vector.

Invoking (13) together with the inverse transformation 𝑣=∫𝑢𝑑𝑥 yields the following independent conservation laws of the (3+1)-dimensional Zakharov-Kuznetsov equation: 𝑇₁=𝑇^𝑡₁,𝑇^𝑥₁,𝑇^𝑦₁,𝑇^𝑧₁,𝑐₁=1 (29) in which 𝑇^𝑡₁=𝑎𝑡𝑢³−𝑥𝑢²−𝑢𝑢𝑑𝑥+𝑦𝑢_𝑦𝑑𝑥+𝑧𝑢_𝑧𝑑𝑥−3𝑡𝑢_𝑥𝑏𝑢_𝑥+𝑢_𝑦𝑦𝑑𝑥+𝑢_𝑧𝑧𝑇𝑑𝑥_𝑥1=−𝑎𝑢²+2𝑏𝑢_𝑥𝑥+𝑢_𝑦𝑦+𝑢_𝑧𝑧+𝑢_𝑡×𝑢𝑑𝑥𝑥𝑢+𝑢𝑑𝑥+3𝑡_𝑡𝑑𝑥+𝑦𝑢_𝑦𝑑𝑥+𝑧𝑢_𝑧+𝑑𝑥2𝑢+3𝑡𝑢_𝑡+𝑥𝑢_𝑥+𝑦𝑢_𝑦+𝑧𝑢_𝑧×2𝑏𝑢_𝑥+𝑢_𝑦𝑦𝑑𝑥+𝑢_𝑧𝑧1𝑑𝑥+𝑥3𝑎𝑢³+𝑢𝑢_𝑡𝑑𝑥−𝑢_𝑥×𝑏𝑢_𝑥+𝑢_𝑦𝑦𝑑𝑥+𝑢_𝑧𝑧𝑇𝑑𝑥_𝑦11=𝑦3𝑎𝑢³−𝑢_𝑥𝑏𝑢+𝑢_𝑦𝑦𝑑𝑥+𝑢_𝑧𝑧𝑑𝑥+𝑢𝑢_𝑡𝑑𝑥−𝑢_𝑥𝑦𝑥𝑢+𝑢𝑑𝑥+3𝑡𝑢_𝑡𝑑𝑥+𝑦𝑢_𝑦𝑑𝑥+𝑧𝑢_𝑧𝑑𝑥+𝑢_𝑥𝑥𝑢_𝑦+2𝑢_𝑦𝑑𝑥+3𝑡𝑢_𝑡𝑦𝑢𝑑𝑥+𝑦_𝑦𝑦𝑑𝑥+𝑧𝑢_𝑦𝑧𝑇𝑑𝑥_𝑧11=𝑧3𝑎𝑢³−𝑢_𝑥𝑏𝑢_𝑥+𝑢_𝑦𝑦𝑑𝑥+𝑢_𝑧𝑧𝑑𝑥+𝑢𝑢_𝑡𝑑𝑥−𝑢_𝑥𝑧𝑥𝑢+𝑢𝑑𝑥+3𝑡𝑢_𝑡𝑑𝑥+𝑦𝑢_𝑦𝑑𝑥+𝑧𝑢_𝑧𝑑𝑥+𝑢_𝑥𝑥𝑢_𝑧+2𝑢_𝑧𝑑𝑥+3𝑡𝑢_𝑡𝑧𝑢𝑑𝑥+𝑦_𝑦𝑧𝑑𝑥+𝑧𝑢_𝑧𝑧𝑇𝑑𝑥₂=𝑢𝑦𝑢_𝑧𝑑𝑥−𝑧𝑢_𝑦𝑑𝑥,𝑇^𝑥₂,𝑇^𝑦₂,𝑇^𝑧₂,𝑐₂=1 (30) in which 𝑇^𝑥₂=𝑎𝑢²+2𝑏𝑢_𝑥𝑥+𝑢_𝑦𝑦+𝑢_𝑧𝑧+𝑢_𝑡×𝑑𝑥𝑦𝑢_𝑧𝑑𝑥−𝑧𝑢_𝑦−𝑑𝑥𝑦𝑢_𝑧−𝑧𝑢_𝑦2𝑏𝑢_𝑥+𝑢_𝑦𝑦𝑑𝑥+𝑢_𝑧𝑧𝑇𝑑𝑥_𝑦2=13𝑎𝑧𝑢³+𝑧𝑢𝑢_𝑡𝑑𝑥+𝑢_𝑥𝑦𝑦𝑢_𝑧𝑑𝑥−𝑧𝑢_𝑦𝑑𝑥−𝑢_𝑥𝑏𝑧𝑢_𝑥+𝑢_𝑧𝑑𝑥+𝑦𝑢_𝑦𝑧𝑑𝑥+𝑧𝑢_𝑧𝑧𝑇𝑑𝑥_𝑧21=−3𝑎𝑦𝑢³−𝑦𝑢𝑢_𝑡𝑑𝑥+𝑢_𝑥𝑧𝑦𝑢_𝑧𝑑𝑥−𝑧𝑢_𝑦𝑑𝑥+𝑢_𝑥𝑏𝑦𝑢_𝑥+𝑢_𝑦𝑑𝑥+𝑧𝑢_𝑦𝑧𝑑𝑥+𝑦𝑢_𝑦𝑦𝑇𝑑𝑥₃=−𝑢𝑢_𝑦𝑑𝑥,𝑇^𝑥₃,𝑇^𝑦₃,𝑢_𝑥𝑢_𝑦𝑧𝑑𝑥−𝑢_𝑥𝑧𝑢_𝑦,𝑐𝑑𝑥₃=1 (31) in which 𝑇^𝑥₃=𝑢_𝑦2𝑏𝑢_𝑥+𝑢_𝑦𝑦𝑑𝑥+𝑢_𝑧𝑧−𝑑𝑥𝑎𝑢²+2𝑏𝑢_𝑥𝑥+𝑢_𝑦𝑦+𝑢_𝑧𝑧+𝑢_𝑡𝑑𝑥𝑢_𝑦𝑇𝑑𝑥_𝑦3=13𝑎𝑢³+𝑢𝑢_𝑡𝑑𝑥−𝑢_𝑥𝑏𝑢_𝑥+𝑢_𝑧𝑧𝑑𝑥−𝑢_𝑥𝑦𝑢_𝑦𝑇𝑑𝑥₄=−𝑢𝑢_𝑧𝑑𝑥,𝑇^𝑥₄,𝑢_𝑥𝑢_𝑦𝑧𝑑𝑥−𝑢_𝑥𝑦𝑢_𝑧𝑑𝑥,𝑇^𝑧₄,𝑐₄=1 (32) in which 𝑇^𝑥₄=𝑢_𝑧2𝑏𝑢_𝑥+𝑢_𝑦𝑦𝑑𝑥+𝑢_𝑧𝑧−𝑑𝑥𝑎𝑢²+2𝑏𝑢_𝑥𝑥+𝑢_𝑦𝑦+𝑢_𝑧𝑧+𝑢_𝑡𝑑𝑥𝑢_𝑧𝑇𝑑𝑥_𝑧4=13𝑎𝑢³+𝑢𝑢_𝑡𝑑𝑥−𝑢_𝑥𝑏𝑢_𝑥+𝑢_𝑦𝑦𝑑𝑥−𝑢_𝑥𝑧𝑢_𝑧𝑇𝑑𝑥₅=𝑇^𝑡₅,𝑇^𝑥₅,𝑢_𝑥𝑢_𝑡𝑦𝑑𝑥−𝑢_𝑥𝑦𝑢_𝑡𝑑𝑥,𝑢_𝑥𝑢_𝑡𝑧𝑑𝑥−𝑢_𝑥𝑧𝑢_𝑡,𝑐𝑑𝑥₅=1 (33) in which 𝑇^𝑡₅=13𝑎𝑢³−𝑢_𝑥𝑏𝑢_𝑥+𝑢_𝑦𝑦𝑑𝑥+𝑢_𝑧𝑧𝑇𝑑𝑥_𝑥5=𝑢_𝑡2𝑏𝑢_𝑥+𝑢_𝑦𝑦𝑑𝑥+𝑢_𝑧𝑧−𝑑𝑥𝑎𝑢²+2𝑏𝑢_𝑥𝑥+𝑢_𝑦𝑦+𝑢_𝑧𝑧+𝑢_𝑡𝑑𝑥𝑢_𝑡𝑇𝑑𝑥₆=𝑥𝑢−𝑎𝑡𝑢²−𝑢𝑑𝑥,𝑇^𝑥₆,𝑎𝑡𝑢_𝑥𝑢_𝑦+(𝑥−𝑎𝑡𝑢)𝑢_𝑥𝑦,𝑎𝑡𝑢_𝑥𝑢_𝑧+(𝑥−𝑎𝑡𝑢)𝑢_𝑥𝑧,𝑐₆=1 (34) in which 𝑇^𝑥₆=𝑥𝑎𝑢²+2𝑏𝑢_𝑥𝑥+𝑢_𝑦𝑦+𝑢_𝑧𝑧+𝑢_𝑡𝑑𝑥−𝑢_𝑦𝑦𝑑𝑥−𝑢_𝑧𝑧2𝑑𝑥−𝑎𝑡3𝑎𝑢³+2𝑏𝑢𝑢_𝑥𝑥+𝑢𝑢_𝑦𝑦+𝑢𝑢_𝑧𝑧−𝑏𝑢²_𝑥𝑇₇=−𝑢²,𝑇^𝑥₇,𝑢_𝑥𝑢_𝑦−𝑢𝑢_𝑥𝑦,𝑢_𝑥𝑢_𝑧−𝑢𝑢_𝑥𝑧𝑐₇=1 (35) in which 𝑇^𝑥₇=𝑏𝑢²_𝑥−23𝑎𝑢³−𝑢2𝑏𝑢_𝑥𝑥+𝑢_𝑦𝑦+𝑢_𝑧𝑧𝑇₈=𝛼+𝛽𝑢,𝑇^𝑥₈,𝛼+𝛽𝑢_𝑥𝑦−𝑖𝛼₂−𝛽₂𝑢_𝑥,𝛼+𝛽𝑢_𝑥𝑧−𝛼₂+𝛽₂𝑢_𝑥 (36) in which 𝑇^𝑥₈=𝛼+𝛽𝑎𝑢²+2𝑏𝑢_𝑥𝑥+𝑢_𝑦𝑦+𝑢_𝑧𝑧+𝑢_𝑡−𝛼𝑑𝑥₁+𝛽₁𝑢𝑑𝑥 (37) and 𝛼₁,  𝛼₂ are the derivatives of 𝛼(𝑡,𝑧−𝑦𝑖) with respect to the first and second gauge terms, respectively. The similiar case is 𝛽₁,  𝛽₂.

Note that (35) is a local conservation law, while the others are nonlocal conservation laws. We have verified that there exist infinite many conservation laws for the Zakharov-Kuznetsov equation.

4. Double Reduction of the Zakharov-Kuznetsov Equation via (35)

In this section, the generalized double reduction method will be applied to the Zakharov-Kuznetsov equation by using the obtained local conservation law (35) and its associated symmetries.

In the paper [13], the authors give out the following seven infinitesimal generators of this equation: 𝑋₁=𝜕𝜕𝑥,𝑋₂=𝜕𝜕𝑦,𝑋₃=𝜕𝜕𝑧,𝑋₄=𝜕,𝑋𝜕𝑡₅𝜕=𝑧𝜕𝜕𝑦−𝑦𝜕𝑧,𝑋₆𝜕=𝑎𝑡+𝜕𝜕𝑥,𝑋𝜕𝑢₇𝜕=𝑥𝜕𝜕𝑥+𝑦𝜕𝜕𝑦+𝑧𝜕𝜕𝑧+3𝑡𝜕𝜕𝑡−2𝑢.𝜕𝑢 (38) We first determine which of these seven symmetries are associated with the conservation law (35) through the following formula: 𝑋𝑇^𝑡𝑇^𝑥𝑇^𝑦𝑇^𝑧−𝐷_𝑡𝜉^𝑡𝐷_𝑥𝜉^𝑡𝐷_𝑦𝜉^𝑡𝐷_𝑧𝜉^𝑡𝐷_𝑡𝜉^𝑥𝐷_𝑥𝜉^𝑥𝐷_𝑦𝜉^𝑥𝐷_𝑧𝜉^𝑥𝐷_𝑡𝜉^𝑦𝐷_𝑥𝜉^𝑦𝐷_𝑦𝜉^𝑦𝐷_𝑧𝜉^𝑦𝐷_𝑡𝜉^𝑧𝐷_𝑥𝜉^𝑧𝐷_𝑦𝜉^𝑧𝐷_𝑧𝜉^𝑧𝑇^𝑡𝑇^𝑥𝑇^𝑦𝑇^𝑧+𝐷_𝑡𝜉^𝑡+𝐷_𝑥𝜉^𝑥+𝐷_𝑦𝜉^𝑦+𝐷_𝑧𝜉^𝑧𝑇^𝑡𝑇^𝑥𝑇^𝑦𝑇^𝑧=0. (39) It can be shown that the associated symmetries are 𝑋₁,𝑋₂,𝑋₃,𝑋₄, and 𝑋₅, but for simplicity we adopt the combination of 𝑋=𝑋₄+𝑐₁𝑋₁+𝑐₂𝑋₂+𝑐₃𝑋₃=𝜕/𝜕𝑡+𝑐₁(𝜕/𝜕𝑥)+𝑐₂(𝜕/𝜕𝑦)+𝑐₃(𝜕/𝜕𝑧) to reduce the conserved vector, in which 𝑐₁,𝑐₂, and 𝑐₃ are three constants, where the generator 𝑋 has a canonical form 𝑋=𝜕/𝜕𝑞 if 𝑑𝑡1=𝑑𝑥𝑐₁=𝑑𝑦𝑐₂=𝑑𝑧𝑐₃=𝑑𝑢0=𝑑𝑟0=𝑑𝑠0=𝑑𝑝0=𝑑𝑞1=𝑑𝑤0 (40) or 𝑟=𝑥−𝑐₁𝑡,𝑠=𝑦−𝑐₂𝑡,𝑝=𝑧−𝑐₃𝑡,𝑞=𝑡,𝑤𝑟,𝑠,𝑝=𝑢. (41) Using the following formula, we can achieve the reduced conserved vector 𝑇^𝑟𝑇^𝑠𝑇^𝑝𝑇^𝑞=𝐽𝐴^−1𝑇𝑇^𝑡𝑇^𝑥𝑇^𝑦𝑇^𝑧, (42) in which 𝐴⁻¹=𝐷_𝑡𝑟𝐷_𝑡𝑠𝐷_𝑡𝑝𝐷_𝑡𝑞𝐷_𝑥𝑟𝐷_𝑥𝑠𝐷_𝑥𝑝𝐷_𝑥𝑞𝐷_𝑦𝑟𝐷_𝑦𝑠𝐷_𝑦𝑝𝐷_𝑦𝑞𝐷_𝑧𝑟𝐷_𝑧𝑠𝐷_𝑧𝑝𝐷_𝑧𝑞,𝐽=det(𝐴). (43) Then the reduced conserved vector is 𝐷_𝑟𝑇^𝑟+𝐷_𝑠𝑇^𝑠+𝐷_𝑝𝑇^𝑝=0, (44) where 𝑇^𝑟=−𝑐₁𝑤²+23𝑎𝑤³−𝑏𝑤²_𝑟+𝑤2𝑏𝑤_𝑟𝑟+𝑤_𝑠𝑠+𝑤_𝑝𝑝,𝑇^𝑠=𝑤𝑤_𝑟𝑠−𝑤_𝑟𝑤_𝑠−𝑐₂𝑤²,𝑇^𝑝=𝑤𝑤_𝑟𝑝−𝑤_𝑟𝑤_𝑝−𝑐₃𝑤²,𝑇^𝑞=𝑤². (45) Similarly, we can determine the associated symmetry with the reduced conserved vector (44) through the following formula: 𝑌𝑇^𝑟𝑇^𝑠𝑇^𝑝−𝐷_𝑟𝜉^𝑟𝐷_𝑠𝜉^𝑟𝐷_𝑝𝜉^𝑟𝐷_𝑟𝜉^𝑠𝐷_𝑠𝜉^𝑠𝐷_𝑝𝜉^𝑠𝐷_𝑟𝜉^𝑝𝐷_𝑠𝜉^𝑝𝐷_𝑝𝜉^𝑝𝑇^𝑟𝑇^𝑠𝑇^𝑝+𝐷_𝑟𝜉^𝑟+𝐷_𝑠𝜉^𝑠+𝐷_𝑝𝜉^𝑝𝑇^𝑟𝑇^𝑠𝑇^𝑝=0. (46) One can verify that the associated symmetries are 𝑌₁=𝜕𝜕𝑟,𝑌₂=𝜕𝜕𝑠,𝑌₃=𝜕.𝜕𝑝 (47) So we can get further reduced conserved vector by the combination of 𝑌=𝑌₁+𝑑₁𝑌₂+𝑑₂𝑌₃=𝜕/𝜕𝑟+𝑑₁(𝜕/𝜕𝑠)+𝑑₂(𝜕/𝜕𝑝), (𝑑₁,𝑑₂ are two constants), and the generator 𝑌 has a canonical form 𝑌=𝜕/𝜕ℎ if 𝑑𝑟1=𝑑𝑠𝑑₁=𝑑𝑝𝑑₂=𝑑𝑤0=𝑑𝑓0=𝑑𝑔0=𝑑ℎ1=𝑑𝜙0 (48) or 𝑓=𝑠−𝑑₁𝑟,𝑔=𝑝−𝑑₂𝑟,ℎ=𝑟,𝜙𝑓,𝑔=𝑤. (49) Through the following formula, we can get the reduced conserved vector: 𝑇^𝑓𝑇^𝑔𝑇^ℎ=𝐽𝐴^−1𝑇𝑇^𝑟𝑇^𝑠𝑇^𝑞, (50) in which 𝐴⁻¹=𝐷_𝑟𝑓𝐷_𝑟𝑔𝐷_𝑟ℎ𝐷_𝑠𝑓𝐷_𝑠𝑔𝐷_𝑠ℎ𝐷_𝑝𝑓𝐷_𝑝𝑔𝐷_𝑝ℎ,𝐽=det(𝐴). (51) And the reduced conserved vector is 𝐷_𝑓𝑇^𝑓+𝐷_𝑔𝑇^𝑔=0, (52) where 𝑇^𝑓=𝑐₁𝑑₁−𝑐₂𝜙²−23𝑎𝑑₁𝜙³+𝑑₁𝜙_𝑓+𝑑₂𝜙_𝑔×𝑏𝑑₁𝑑₂𝜙_𝑔+1+𝑏𝑑²₁𝜙_𝑓−𝜙𝑑₁1+2𝑏𝑑²₂𝜙_𝑔𝑔+𝑑₂1+4𝑏𝑑²₁𝜙_𝑓𝑔+2𝑑₁1+𝑏𝑑²₁𝜙_𝑓𝑓,𝑇^𝑔=𝑐₁𝑑₂−𝑐₃𝜙²−23𝑎𝑑₂𝜙³+𝑑₁𝜙_𝑓+𝑑₂𝜙_𝑔×𝑏𝑑₁𝑑₂𝜙_𝑓+1+𝑏𝑑²₂𝜙_𝑔−𝜙𝑑₂1+2𝑏𝑑²₁𝜙_𝑓𝑓+𝑑₁1+4𝑏𝑑²₂𝜙_𝑓𝑔+2𝑑₂1+𝑏𝑑²₂𝜙_𝑔𝑔,𝑇^ℎ=−𝑐₁𝜙²+23𝑎𝜙³−𝑏𝑑₁𝜙_𝑓+𝑑₂𝜙_𝑔²+𝜙1+2𝑏𝑑²₁𝜙_𝑓𝑓+4𝑏𝑑₁𝑑₂𝜙_𝑓𝑔+1+2𝑏𝑑²₂𝜙_𝑔𝑔. (53)

Similarly, one can determine the associated symmetry with the reduced conserved vector (52) through the same formula, here we omit. One can verify that the associated symmetries with (52) are 𝑍₁=𝜕𝜕𝑓,𝑍₂=𝜕.𝜕𝑔 (54)

So we can get further reduced conserved vector by the combination of 𝑍=𝑍₁+𝑒₁𝑍₂=𝜕/𝜕𝑓+𝑒₁(𝜕/𝜕𝑔), (𝑒₁ is a constant), and the generator 𝑍 has a canonical form 𝑍=𝜕/𝜕𝜆 if 𝑑𝑓1=𝑑𝑔𝑒₁=𝑑𝜙0=𝑑𝜃0=𝑑𝜆1=𝑑𝜓0 (55) or 𝜃=𝑔−𝑒₁𝑓,𝜆=𝑓,𝜓(𝜃)=𝜙. (56) We can achieve the reduced conserved vector as follows: 𝐷_𝜃𝑇^𝜃+𝐷_𝜆𝑇^𝜆=0, (57) in which 𝑇^𝜃=𝑐₃−𝑐₂𝑒₁−𝑑₂−𝑑₁𝑒₁𝑐₁𝜓²+23𝑎𝑑₂−𝑑₁𝑒₁𝜓³−𝑑₂−𝑑₁𝑒₁1+𝑒²₁+𝑏𝑑₂−𝑑₁𝑒₁²𝜓^′2+2𝑑₂−𝑑₁𝑒₁1+𝑒²₁+𝑏𝑑₂−𝑑₁𝑒₁²𝜓𝜓^′′,𝑇^𝜆=𝑐₂−𝑐₁𝑑₁𝜓²+23𝑎𝑑₁𝜓³+𝑑₂−𝑑₁𝑒₁×𝑒₁−𝑏𝑑₁𝑑₂−𝑑₁𝑒₁𝜓^′2+1+2𝑏𝑑²₂𝑑₁−1+4𝑏𝑑²₁𝑑₂𝑒₁+21+𝑏𝑑²₁𝑑₁𝑒²₁𝜓𝜓^′′. (58)

The last step of double reduction gives the following equation: 2𝐴𝜓𝜓^′′−𝐴𝜓^′2+𝐵𝜓³+𝐶𝜓²=𝐷. (59) It is a second order nonlinear ODE, in which 𝐴=𝑑₂−𝑑₁𝑒₁1+𝑒²₁+𝑏𝑑₂−𝑑₁𝑒₁²,2𝐵=3𝑎𝑑₂−𝑑₁𝑒₁,𝐶=𝑐₃−𝑐₂𝑒₁−𝑑₂−𝑑₁𝑒₁𝑐₁, (60) and 𝐷 is a constant.

It also has 𝑢(𝑡,𝑥,𝑦,𝑧)=𝜓(𝜃), 𝜃=((𝑑₂−𝑑₁𝑒₁)𝑐₁−𝑐₃+𝑐₂𝑒₁)𝑡−(𝑑₂−𝑑₁𝑒₁)𝑥−𝑒₁𝑦+𝑧.

5. Exact Solutions for the Zakharov-Kuznetsov Equation

If we have solved the obtained nonlinear ODE (59), then exact solutions for the Zakharov-Kuznetsov equation can be obtained after substituting the original variables.

Note that (59) can be rewritten as 𝜓^′′=𝜓^′2−𝐵2𝜓𝜓2𝐴²−𝐶𝐷2𝐴𝜓+12𝐴𝜓. (61) Under the transformation 𝜓^′=𝜒(𝜓), (55) becomes 𝜒^′𝜓=1𝐷2𝜓𝜒+12𝐴𝜓−𝐵𝜓2𝐴²−𝐶𝜓𝜒2𝐴⁻¹, (62) which is a Bernoulli equation. Its general solution is given by 𝜒²𝜓𝐷=−𝐴𝐶+𝐾𝜓−𝐴𝜓²−𝐵𝜓2𝐴³ (63) in which 𝐾 is an integral constant.

Recall that 𝜓^′=𝜒(𝜓), so we have 𝜓^′(𝜃)=±−𝐷𝐴𝐶+𝐾𝜓−𝐴𝜓²−𝐵𝜓2𝐴³. (64)

Note that Fan once obtained some exact solutions for the following more general nonlinear ODE: 𝑑𝜙(𝜉)𝑑𝜉=𝜀⁴_𝑖=0𝑐_𝑖𝜙^𝑖, (65) where 𝜀=±1,  𝑐_𝑖(𝑖=0,1,2,3,4) are constants.

For this knowledge, we refer readers to [15]. But for volumes, they are not presented here.

Next, exact solutions of (64) will be constructed with the aid of the obtained results by Fan, and thus they are also solutions of the Zakharov-Kuznetsov equation after substituting the original variables.

Two particular cases will be investigated as follows.

Case 1 (𝐷=𝐾=0). This case can be further divided into three subclasses as follows.(1)If 𝐶/𝐴<0, we have 𝑢𝑡,𝑥,𝑦,𝑧=−2𝐶𝐵secℎ²12−𝐶𝐴𝜃. (66)(2)If 𝐶/𝐴>0, we have 𝑢𝑡,𝑥,𝑦,𝑧=−2𝐶𝐵sec²12𝐶𝐴𝜃. (67)(3)If 𝐶=0, we have 𝑢𝑡,𝑥,𝑦,𝑧=−2𝐴𝐵𝜃². (68)Which are the bell-shape soliton solution, the periodic triangular function solution and the rational function solution, respectively.

Case 2 (𝐵/𝐴<0). In this case, we achieve a solution 𝑢1𝑡,𝑥,𝑦,𝑧=℘2−𝐵2𝐴𝜃,𝑔₂,𝑔₃, (69) in which 𝑔₂=8𝐴𝐾/𝐵,𝑔₃=16𝐷/𝐵. This a Weierstrass elliptic function solution.

In all of the obtained solutions, 𝐴=(𝑑₂−𝑑₁𝑒₁)(1+𝑒²₁+𝑏(𝑑₂−𝑑₁𝑒₁)²),   𝐵=(2/3)𝑎(𝑑₂−𝑑₁𝑒₁),   𝐶=𝑐₃−𝑐₂𝑒₁−(𝑑₂−𝑑₁𝑒₁)𝑐₁,  and 𝜃=((𝑑₂−𝑑₁𝑒₁)𝑐₁−𝑐₃+𝑐₂𝑒₁)𝑡−(𝑑₂−𝑑₁𝑒₁)𝑥−𝑒₁𝑦+𝑧.

Remark 8. If we substitute (64) into (61) and eliminate 𝜓^′ in (61), then we can get the following equation: 𝜓^′′(𝜃)=𝐾2−𝐶𝐴𝜓−3𝐵𝜓4𝐴². (70) Solving (70) can also help us find solutions to the Zakharov-Kuznetsov equation, which will be discussed in our future research paper.

6. Conclusion

In this work, in order to apply the Noether’s approach to construct conservation laws for the (3+1)-dimensional Zakharov-Kuznetsov equation, the transformation 𝑢=𝑣_𝑥 is utilized. It is of interest to find that a standard Lagrangian exists for the transformed equation. Then Noether’s approach is used to derive the conservation laws. Finally, conservation laws for the Zakharov-Kuznetsov equation are obtained under the inverse transformation 𝑣=∫𝑢𝑑𝑥, which involves local and nonlocal conservation laws. Many infinite conservation laws of this equation are also obtained, which shows the strong integrability of this equation. Then, with the aid of the local conservation law, the generalized double reduction theorem is applied to the Zakharov-Kuznetsov equation. It has been shown that the reduced equation is a second order nonlinear ODE. Some exact solutions of the Zakharov-Kuznetsov equation are obtained after solving the nonlinear ODE. In addition, Xia and Xiong [16] took an interesting approach to investigate soliton solutions of nonlinear equations, which induces us to consider some exact solutions of (1).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.