# The First-Integral Method and Abundant Explicit Exact Solutions to the Zakharov Equations

Authors: Yadong Shang; Xiaoxiao Zheng
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper is concerned with the system of Zakharov equations which involves the interactions between Langmuir and ion-acoustic waves in plasma. Abundant explicit and exact solutions of the system of Zakharov equations are derived uniformly by using the first integral method. These exact solutions are include that of the solitary wave solutions of bell-type forn and E, the solitary wave solutions of kink-type for E and bell-type for n, the singular traveling wave solutions, periodic wave solutions of triangle functions, Jacobi elliptic function doubly periodic solutions, and Weierstrass elliptic function doubly periodic wave solutions. The results obtained confirm that the first integral method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial differential equations.


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## Body

## 1. Introduction

Zakharov equations(1.1)ntt-cs2nxx=β(|E|2)xx,iEt+αExx=δnE
have been presented by Zakharov and others in 1972 [1, 2] to model the interactions of laser-plasma. In (1.1), n is the perturbed number density of the ion (in the low frequency response); E is the slow variation amplitude of the electric field intensity; cs is the thermal transportation velocity of the electron-ion; α≠0, β≠0, δ≠0, cs are constants. Equations (1.1) are one of the fundamental models governing dynamics of nonlinear waves and describing the interactions between high- and low-frequency waves. In the interaction of laser-plasma the system of Zakharov equations plays an important role (see [3] and references therein).More recently, some authors considered the exact and explicit solutions of the system of Zakharov equations by different methods in [4–8]. In [3], the author established the traveling wave solutions for (1.1) by analytical method. The extended hyperbolic function method was employed to find some solitary wave solutions for (1.1) in  [4, 5]. In  [6, 7], the authors obtained elliptic function solutions for (1.1) by the Jacobi elliptic function method and the generalized Jacobi elliptic function expansion method. In  [8], some new traveling wave solutions for (1.1) are obtained by using bifunction method and Wu-elimination method. In the above references the authors can obtain only one type of exact solution by one method.The aim of this paper is to supply a unified method for constructing a series of explicit exact solutions to the system of Zakharov equations (1.1). The first integral method is employed to investigate the system of Zakharov equations (1.1). Through an exhaustive analysis and discussion for different parameters, we uniformly construct a series of explicit exact solutions to (1.1). Compared with most methods used in [4–8] such as the extended hyperbolic function method, Jacobi elliptic function method, and its extension, the first integral method not only gives abundant explicit exact solitary wave solutions, periodic wave solutions of triangle function, but also provides Jacobi elliptic function and Weierstrass elliptic function doubly periodic wave solutions.The rest of this paper is organized as follows. In Section2, the outline of the first integral method will be given. Section 3 is the main part of this paper; the method is employed to seek the explicit and exact solutions of the system of Zakharov equations (1.1). In the last section, some conclusion is given.


## 2. The First Integral Method

The first-integral method, which is based on the ring theory of commutative algebra, was first proposed by Prof. Feng Zhaosheng [9] in 2002. The method has been applied by Feng to solve Burgers-KdV equation, the compound Burgers-KdV equation, an approximate Sine-Gordon equation in (n+1)-dimensional space, and two-dimensional Burgers-KdV equation [10–14].Recent years, many authors employed this method to solve different types of nonlinear partial differential equations in physical mathematics. More information about these applications can be found in [15] and references therein. The most advantage is that the first integral method does not have many sophisticated computation in solving nonlinear algebra equations compared to other direct algebra methods. For the sake of completeness, we briefly outline the main steps of this method.The main steps of this method are summarized as follows.Given a system of nonlinear partial differential  equations, for example, in two independent variables(2.1)P(ut,ux,uxx,uxt,…)=0,Q(vt,vx,vxx,vxt,…)=0,
and using traveling wave transformation u(x,t)=f(ξ),  v(x,t)=g(ξ),  ξ=kx+ωt+ξ0 and some other mathematical operations, the systems (2.1) can be reduced to a second order nonlinear ordinary differential equation
(2.2)D(f,f′,f′′)=0.
By introducing new variables X=f(ξ),  Y=f′(ξ), or making some other transformations we reduce ordinary differential equation (2.2) to a system of the first order ordinary differential equation
(2.3)X′=Y,Y′=H(X,Y).
Suppose that the first integral of (2.3) has a form as follows:
(2.4)P(X,Y)=∑i=om‍ai(X)Yi=0
(in general m=1 or m=2), where ai (i=0,1,…,m) are real polynomials of X.According to the Division theorem there exists polynomialsα(X),  β(X) of variable X in ℜ[X] such that
(2.5)dPdξ=[α(X)+β(X)Y]P(X,Y).We determine polynomialsα(X), β(X), ai(X) (i=0,1,2,…) from (2.5) and, furthermore, obtain P(X,Y).Then substitutingX=f(ξ),  Y=f′(ξ) or other transformations into (2.4), exact solutions to (2.1) is established, through solving the resulting first order integrable differential equation.


## 3. Explicit and Exact Solutions of the System of Zakharov Equations

In this section we will employ the first integral method to construct abundant explicit exact traveling wave solutions to (1.1).In order to transfer (1.1) into the form of (2.2), we firstly do some transformations for (1.1). Since E(x,t) in (1.1) is a complex function and we are seeking for the traveling wave solutions, we introduce a gauge transformation
(3.1)E(x,t)=ei(kx+ωt+ξ0)φ(x,t),
where φ(x,t) is real-valued function, k, ω are two real constants to be determined later, and ξ0 is an arbitrary constant. Substituting (3.1) into (1.1), we have
(3.2)ntt-cs2nxx=β(φ2)xx,(3.3)φt+2αkφx=0,(3.4)αφxx-(αk2+ω)φ=δnφ.
In the view of (3.3), we suppose
(3.5)φ(x,t)=φ(ξ)=φ(x-2αkt+ξ1),
where ξ1 is an arbitrary constant. Substituting (3.5) into (3.4), we infer that
(3.6)n(x,t)=αφxx-(ω+αk2)φδφ=αφ′′(ξ)δφ(ξ)-(ω+αk2).
Therefore, we can also assume
(3.7)n(x,t)=ψ(ξ)=ψ(x-2αkt+ξ1).
Substituting (3.7) into (3.2), and integrating the resultant equation twice with respect to ξ, we obtain
(3.8)(4α2k2-cs2)ψ(ξ)=βφ2(ξ)+C′ξ+C,
where C′,  C are two arbitrary integration constants. For k=±cs/2α, (1.1) have one set of solution
(3.9)E0(x,t)=±[C′(x±cst+ξ0)+C-β]1/2e[i(±(cs/2α)x+ωt+ξ0)],n0(x,t)=-α(C′)24δ[C′(x±cst+ξ0)+C]2-(ω+cs24α).
For k≠±cs/2α, we put C′=0 in (3.8) for the cause of technical. Thus (3.8) becomes
(3.10)ψ(ξ)=β4α2k2-cs2φ2(ξ)+C.
Substituting (3.5), (3.7), and (3.10) into (3.4), we obtain
(3.11)φ′′(ξ)-(ω+αk2+Cδ)αφ(ξ)-δβα(4α2k2-cs2)φ3(ξ)=0.
Let l=(ω+αk2+Cδ)/α,  m=δβ/α(4α2k2-cs2); thus (3.11) becomes the Liénard equation
(3.12)φ′′(ξ)-lφ(ξ)-mφ3(ξ)=0.
Let X=φ(ξ),  Y=X′, (3.12) can be converted to a system of nonlinear ODEs as follows:
(3.13)X′=Y,Y′=lX+mX3.
Now the Division theorem is applied to seek the first integral to (3.13). Suppose that X=X(ξ),  Y=Y(ξ) are the nontrivial solution to the system (3.13), and its first integral is an irreducible polynomial in ℜ[X,Y](3.14)P(X(ξ),Y(ξ))=∑i=0m‍ai(X)Yi=0,
where ai,  i=0,1,2,…,m are polynomials of X. According to the Division theorem, there exists a polynomial T(X,Y)=α(X)+β(X)Y, such that
(3.15)dPdξ=∂P∂X∂X∂ξ+∂P∂Y∂Y∂ξ=[α(X)+β(X)Y](∑i=0m‍ai(X)Yi).Here we only consider the case ofm=2.Substituting (3.13) and (3.14) into (3.15), one gets
(3.16)dPdξ=a0′(X)Y+a1′(X)Y2+[a1(X)+2a2(X)Y]Y′+a2′(x)Y3=(α(X)+β(X)Y)(a0(X)+a1(X)Y+a2(X)Y2).
Collecting all the terms with the same power of Y together and equating each coefficient to zero yields a set of nonlinear algebraic equations as follows:
(3.17)a2′(X)=β(X)a2(X),(3.18)a1′(X)=β(X)a1(X)+α(X)a2(X),(3.19)a0′(X)=β(X)a0(X)+α(X)a1(X)-2a2(X)(lX+mX3),(3.20)a1(X)(lX+mX3)=α(X)a0(X).
Because ai(X) (i=0,1,2) are polynomials, from (3.17) we can deduce deg[a2(X)]=0, β(X)=0; that is, a2(X) is a constant. For simplicity, we take β(X)=0,  a2(X)=1. Then we determine a0(X),  a1(X), and α(X). From (3.18), we have deg[a1(X)]-1=deg[α(X)] or a1(X)=0,  α(X)=0. In what follows we will discuss these two situations.(a) In the case ofa1(X)=0,  α(X)=0.In this case, (3.18) and (3.20) are satisfied. From (3.19), we can derive a0(X)=-(m/2)X4-lX2-d, where d is an integral constant. Substituting a2(X),  a1(X),  and a0(X) into (3.14), one obtain that
(3.21)Y2=m2X4+lX2+d,
that is,
(3.22)X′=±m2X4+lX2+d.
Based on the discussion for different parameters, we can obtain the solutions of the nonlinear ordinary differential equation (3.22).(1) Ford=0,  (3.22) admits the following five general solutions:
(3.23)X1=±-2lmsec-lξ,l<0,m>0,X2=±-2lmcsc-lξ,l<0,m>0,X3=±2lmcschlξ,lm>0,X4=±-2lmsechlξ,l>0,m<0,X5=±1m/2,l=0.
Combining (3.1), (3.7), (3.10), (3.23), and X=φ(ξ), one can get the following five sets of explicit exact solution to (1.1):
(3.24)E1(x,t)=±2(ω+αk2+Cδ)(4α2k2-cs2)δβsec[-(ω+αk2+Cδ)αξ]eiη,n1(x,t)=-2(ω+αk2+Cδ)δsec2[-ω+αk2+Cδαξ]+C,α(ω+αk2+Cδ)<0,αβδ(4α2k2-cs2)>0,(3.25)E2(x,t)=±2(ω+αk2+Cδ)(4α2k2-cs2)δβcsc[-(ω+αk2+Cδ)αξ]eiη,n2(x,t)=-2(ω+αk2+Cδ)δcsc2[-ω+αk2+Cδαξ]+C,α(ω+αk2+Cδ)<0,αβδ(4α2k2-cs2)>0,(3.26)E3(x,t)=±2(ω+αk2+Cδ)(4α2k2-cs2)δβcsch[ω+αk2+Cδαξ]eiη,n3(x,t)=2(ω+αk2+Cδ)δcsch2[ω+αk2+Cδαξ]+C,α(ω+αk2+Cδ)>0,αβδ(4α2k2-cs2)>0,(3.27)E4(x,t)=±2(ω+αk2+Cδ)(4α2k2-cs2)δβsech[ω+αk2+Cδαξ]eiη,n4(x,t)=-2(ω+αk2+Cδ)δsech2[ω+αk2+Cδαξ]+C,α(ω+αk2+Cδ)>0,αβδ(4α2k2-cs2)<0,
where ξ=x-2αkt+ξ1,  η=kx+ωt+η0, k≠±cs/2α, ω, C are arbitrary parameters, and ξ1, η0 are two arbitrary constants. One has
(3.28)E5(x,t)=±2α(4α2k2-cs2)δβ1(x-2αkt+ξ1)eiη,n5(x,t)=2αδ[1x-2αkt+ξ1]2-ω+αk2δ,αδβ(4α2k2-cs2)>0,
where η=kx+ωt+η0, k≠±cs/2α, k,  ω are arbitrary parameters, and ξ1,  η0 are two arbitrary constants.Remark 3.1.
In the above solutions, solutionsE1, n1 (3.24) and E2, n2 (3.25) are explicit exact periodic traveling wave solutions. The solution E3 is an envelope solitary wave solution of bell shape and n3 is a explicit exact solitary wave solutions of bell shape. The n3 be called Langmuir whistler soliton or Langmuir pit soliton according to n3 is positive or negative. The solutions E4, n4 (3.27) are explicit exact singular traveling wave solutions. The singularity will appear as ξ=0 and it represents that the distortions arise from the perturbed number density of the ion n and the electric field intensity E due to instability. While solution E5 (and n5, resp.) (3.28) is an envelope solitary wave solution of bell shape and (a explicit exact solitary wave solutions of bell shape, resp.) in rational function type.(2) Ford=l2/2m, we can obtain following four sets of explicit exact solutions to (3.22)
(3.29)X6=±lmtanl2(ξ+ξ2),l>0,m>0,X7=±lmcotl2(ξ+ξ2),l>0,m>0,X8=±-lmtanh-l2(ξ+ξ2),l<0,m>0,X9=±-lmcoth-l2(ξ+ξ2),l<0,m>0,
where l=(ω+αk2+Cδ)/α,  m=δβ/α(4α2k2-cs2), and  ξ2 is integral constant.Combining (3.1), (3.7), (3.10), (3.29), and X=φ(ξ), we can get the following four explicit exact solution of (1.1):
(3.30)E6(x,t)=±(ω+αk2+Cδ)(4α2k2-cs2)δβtan[ω+αk2+Cδ2αξ]eiη,n6(x,t)=ω+αk2+Cδδtan2[ω+αk2+Cδ2αξ]+C,α(ω+αk2+Cδ)>0,αδβ(4α2k2-cs2)>0,(3.31)E7(x,t)=±(ω+αk2+Cδ)(4α2k2-cs2)δβcot[ω+αk2+Cδ2αξ]eiη,n7(x,t)=ω+αk2+Cδδcot2[ω+αk2+Cδ2αξ]+C,α(ω+αk2+Cδ)>0,αδβ(4α2k2-cs2)>0,(3.32)E8(x,t)=±-(ω+αk2+Cδ)(4α2k2-cs2)δβtanh[-(ω+αk2+Cδ)2αξ]eiη,n8(x,t)=-(ω+αk2+Cδ)δtanh2[-(ω+αk2+Cδ)2αξ]+C,α(ω+αk2+Cδ)<0,αδβ(4α2k2-cs2)>0,(3.33)E9(x,t)=±-(ω+αk2+Cδ)(4α2k2-cs2)δβcoth[-(ω+αk2+Cδ)2αξ]eiη,n9(x,t)=-2(ω+αk2+Cδ)δcoth2[-(ω+αk2+Cδ)2αξ]+C,α(ω+αk2+Cδ)<0,αδβ(4α2k2-cs2)>0,
where ξ=x-2αkt+ξ1,  η=kx+ωt+η0,  k≠±cs/2α,  ω,  C are arbitrary parameters, and ξ1, η0 are two arbitrary constants.Remark 3.2.
The solutionsE6,  n6 (3.30) and E7,  n7 (3.31) are all unbounded periodic traveling wave solutions of triangle function type. The solution E8 is an envelope solitary wave solution of kink type while exact solitary wave solution n8 is dark soliton; that means the density increases as a whole but decreases in part. The exact solutions E9,  n9 (3.32) are explicit exact singular traveling wave solutions. The singularity will be appear as ξ=0 and it indicates that the distortions arise from the perturbed number density of the ion n and the electric field intensity E due to instability.(3) Ford≠0, we obtain elliptic function solutions for (3.22) as follows:
(3.34)X10=±-2(l+1)msnmd2ξ,d=-2(l+1)m,(3.35)X11=±[2d(1+l)m(l-1)]1/4cnl+12ξ,d=l2-12m,(3.36)X12=±2mdnl+12ξ,d=2(l-1)m,(3.37)X13=±2mns-l-1ξ,d=-2(l+1)m,(3.38)X14=±[-l-1m]1/4ncl+12ξ,d=l2-12m,(3.39)X15=±-2(l-1)mnd2-lξ,d=2(l-1)m,(3.40)X16=±2(l-1)msc2-lξ,d=2(l-1)m,(3.41)X17=±l2-12msd2+l2ξ,d=l2-12m,(3.42)X18=±2mcs2-lξ,d=2(l-1)m,(3.43)X19=±-2(l+1)mcd-l-1ξ,d=-2(l+1)m,(3.44)X20=±2mds2+l2ξ,d=l2-12m,(3.45)X21=±2mdc-l-1ξ,d=-2(l+1)m.
Combining (3.1), (3.7), (3.10), the above results (3.33)–(3.44), and X=φ(ξ), we can get the following twelve Jacobi elliptic doubly periodic wave solutions of (1.1)
(3.46)E10(x,t)=±-2(ω+αk2+Cδ+α)(4α2k2-cs2)δβsn(-(ω+αk2+Cδ+α)αξ)eiη,n10(x,t)=-2(ω+αk2+Cδ+α)δsn2(-(ω+αk2+Cδ+α)αξ)+C,d=-2(ω+αk2+Cδ+α)(4α2k2-cs2)δβ,(3.47)E11(x,t)=±(ω+αk2+Cδ+α)(4α2k2-cs2)δβcn(ω+αk2+Cδ+α2αξ)eiη,n11(x,t)=ω+αk2+Cδ+αδcn2(ω+αk2+Cδ+α2αξ)+C,d=[(ω+αk2+Cδ)2-α2](4α2k2-cs2)2αδβ,(3.48)E12(x,t)=±2α(4α2k2-cs2)δβdn(2α-(ω+αk2+Cδ)αξ)eiη,n12(x,t)=2αδdn2(2α-(ω+αk2+Cδ)αξ)+C,d=2(ω+αk2+Cδ-α)(4α2k2-cs2)δβ,(3.49)E13(x,t)=±2α(4α2k2-cs2)δβns(-(ω+αk2+Cδ+α)αξ)eiη,n13(x,t)=2αδns2(-(ω+αk2+Cδ+α)αξ)+C,d=-2(ω+αk2+Cδ+α)(4α2k2-cs2)δβ,(3.50)E14(x,t)=±-(ω+αk2+Cδ-α)(4α2k2-cs2)δβnc(ω+αk2+Cδ+2α2αξ)eiη,n14(x,t)=-ω+αk2+Cδ-αδnc2(ω+αk2+Cδ+2α2αξ)+C,d=[(ω+αk2+Cδ)2-α2](4α2k2-cs2)2αδβ,(3.51)E15(x,t)=±-2(ω+αk2+Cδ-α)(4α2k2-cs2)δβnd(2α-(ω+αk2+Cδ)2αξ)eiη,n15(x,t)=-2(ω+αk2+Cδ-α)δnd2(2α-(ω+αk2+Cδ)2αξ)+C,d=2(ω+αk2+Cδ-α)(4α2k2-cs2)δβ,(3.52)E16(x,t)=±2(ω+αk2+Cδ-α)(4α2k2-cs2)δβsc(2α-(ω+αk2+Cδ)2αξ)eiη,n16(x,t)=2(ω+αk2+Cδ-α)δsc2(2α-(ω+αk2+Cδ)2αξ)+C,d=2(ω+αk2+Cδ-α)(4α2k2-cs2)δβ,(3.53)E17(x,t)=±[(ω+αk2+Cδ)2-α2](4α2k2-cs2)2αδβsd(ω+αk2+Cδ+2α2αξ)eiη,n17(x,t)=(ω+αk2+Cδ)2-α22αδsd2(ω+αk2+Cδ+2α2αξ)+C,d=[(ω+αk2+Cδ)2-α2](4α2k2-cs2)2αδβ,(3.54)E18(x,t)=±2α(4α2k2-cs2)δβcs(2α-(ω+αk2+Cδ)2αξ)eiη,n18(x,t)=2αδcs2(2α-(ω+αk2+Cδ)2αξ)+C,d=2(ω+αk2+Cδ-α)(4α2k2-cs2)δβ,(3.55)E19(x,t)=±-2(ω+αk2+Cδ+α)(4α2k2-cs2)δβcd(-(ω+αk2+Cδ+α)αξ)eiη,n19(x,t)=-2(ω+αk2+Cδ+α)δcd2(-(ω+αk2+Cδ+α)αξ)+C,d=-2(ω+αk2+Cδ+α)(4α2k2-cs2)δβ,(3.56)E20(x,t)=±2α(4α2k2-cs2)δβds(ω+αk2+Cδ+2α2αξ)eiη,n20(x,t)=2αδds2(ω+αk2+Cδ+2α2αξ)+C,d=[(ω+αk2+Cδ)2-α2](4α2k2-cs2)2αδβ,(3.57)E21(x,t)=±2α(4α2k2-cs2)δβdc(-(ω+αk2+Cδ+α)αξ)eiη,n21(x,t)=2αδdc2(-(ω+αk2+Cδ+α)αξ)+C,d=-2(ω+αk2+Cδ+α)(4α2k2-cs2)δβ,
where ξ=x-2αkt+ξ1,  η=kx+ωt+η0,  k≠±cs/2α,  ω,  C are arbitrary parameters, and ξ1,  η0 are two arbitrary constants.(4) Forl=0, m≠0, from (3.22) we have
(3.58)X′=±m2X4+d.
Let X2=Z; (3.58) becomes
(3.59)Z′(ξ)=±2mZ3+4dZ.
While m>0, the above equation possesses a Weierstrass elliptic function doubly periodic wave type solution
(3.60)Z=℘(m2ξ,-8dm,0).
So (3.58) admits a Weierstrass elliptic function doubly periodic wave type solution
(3.61)X22=±℘(m2ξ,-8dm,0).
Combining (3.1), (3.7), (3.10), the above result (3.61), and X=φ(ξ), we derive that (1.1) admits a Weierstrass elliptic function doubly periodic wave type solution
(3.62)E22(x,t)=±℘(δβ2α(4α2k2-cs2)ξ,-8dδβα(4α2k2-cs2),0)ei(kx+ωt+ξ0),n22(x,t)=β4α2k2-cs2℘(δβ2α(4α2k2-cs2)ξ,-8dδβα(4α2k2-cs2),0)+C,
where ξ=x-2αkt+ξ1.Remark 3.3.
The above twelve explicit exact Jacobi elliptic doubly periodic wave solutionsE10,  n10 (3.46); E21,  n21 (3.57); and explicit exact Weierstrass elliptic doubly periodic wave solution E22,  n22 (3.62) have not been obtained in the author’s previous work [4] or other literature [5–8]. It should be emphasized that explicit exact Weierstrass elliptic doubly periodic wave solution E22,  n22 (3.62) is obtained in this paper firstly.(b) In the case ofdeg[a1(X)]-1=deg[α(X)].In this case, we assume thatdeg[α(X)]=k1,  deg[a0(X)]=k2, then we have deg[a1(X)]=k1+1. Now, by balancing the degrees of both sides of (3.20), we can deduce that k2=4. By balancing the degrees of both sides of (3.19), we can also conclude that k1=1 or k1=0. If k1=0 assuming that α(X)=A0,  a1(X)=A1X+A2,  a0(X)=C4X4+C3X3+C2X2+C1X+C0 and substituting them into (3.18)–(3.20); by equating the coefficients of the different powers of X on both sides of (3.18) to (3.20), we can get that α(X)=a1(X)=0. This contradicts with our assumption. It indicates that k1≠0. While k1=1, assuming that a0(X)=C4X4+C3X3+C2X2+C1X+C0,  a1=A2X2+A1X+A0,  α(X)=B1X+B0 then substituting these representations into (3.18)–(3.20), and by equating the coefficients of the different powers of X on both sides of (3.18) to (3.20), we can obtain an overdetermined system of nonlinear algebraic equations
(3.63)2A2=B1,A1=B0,4C4+2m=A2B1,3C3=A1B1+A2B0,2C2+2l=A1B0+A0B1,C1=A2B0,mA2=B1C4,mA1=B1C3+C4B0,mA0+lA2=B1C2+B0C3,lA1=B1C1+B0C2,lA2=B1C0+B0C1,B0C0=0.
By analyzing all kinds of possibilities, we have the following.(1)
WhileB0=C0=0, it leads to a contradiction.(2)
WhileB0≠0,C0=0, it also leads to a contradiction.(3)
WhileB0=0,C0≠0, we can derive that
(3.64)A0=±l2mm,A1=0,A2=±2m,C0=l22m,C1=C3=0,C2=l,C4=m2.Setting (3.64) in (3.14) yields
(3.65)(X′)2±(2mX2+lm2m)X′+m2X4+lX2+l22m=0,
that is,
(3.66)dXX2+l/m=±m2(ξ+ξ0).
Solving (3.66) we can obtain solutions X6,  X7,  X8, and X9 again. Consequently, we obtain explicit exact solutions E6,  n6;  E7,  n7;  E8,  n8;  E9,  n9 to (1.1). Here we will not list them one by one.


## 4. Summary and Conclusions

In summary, we employ the first integral method to uniformly construct a series of explicit exact solutions for a system of Zakharov equations. Abundant explicit exact solutions to Zakharov equations are obtained through an exhaustive analysis and discussion of different parameters. The exact solutions obtained in this paper include that of the solitary wave solutions of bell type forn and E, the solitary wave solutions of kink-type for E and bell type for n, the two kinds of singular traveling wave solutions, four kinds of periodic wave solutions of triangle functions, twelve kinds of Jacobi elliptic function doubly periodic solutions, and one kind of Weierstrass elliptic function doubly periodic wave solutions. Among these are entirely new solutions that first reported in this paper. Some known results of previous references are enriched greatly. The results indicate that the first integral method is a very effective method to solve nonlinear differential equation. The method also is readily applicable to a large variety of other nonlinear evolution equations in physical mathematics. Of course, the first integral method also has its limitations. These solutions are not general and by no means exhaust all possibilities. Such solitary wave solutions of a compound of the bell shape and the kink-shape for n and E established in [4] cannot be obtained by using the first integral method. Some Jacobi elliptic function solutions obtained in [16] also cannot be established in here.



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