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

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 for n 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.

1. Introduction

Zakharov equations 𝑛_𝑡𝑡−𝑐²_𝑠𝑛_𝑥𝑥=𝛽|𝐸|²_𝑥𝑥,𝑖𝐸_𝑡+𝛼𝐸_𝑥𝑥=𝛿𝑛𝐸 (1.1) 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; 𝑐_𝑠 is the thermal transportation velocity of the electron-ion; 𝛼≠0, 𝛽≠0, 𝛿≠0, 𝑐_𝑠 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 Section 2, 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 𝑃𝑢_𝑡,𝑢_𝑥,𝑢_𝑥𝑥,𝑢_𝑥𝑡𝑄𝑣,…=0,_𝑡,𝑣_𝑥,𝑣_𝑥𝑥,𝑣_𝑥𝑡,…=0, (2.1) 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 𝐷𝑓,𝑓^′,𝑓^′′=0. (2.2) 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 𝑋^′𝑌=𝑌,_′=𝐻(𝑋,𝑌). (2.3) Suppose that the first integral of (2.3) has a form as follows: 𝑃(𝑋,𝑌)=^𝑚_𝑖=𝑜𝑎_𝑖(𝑋)𝑌^𝑖=0 (2.4) (in general 𝑚=1 or 𝑚=2), where 𝑎_𝑖 (𝑖=0,1,…,𝑚) are real polynomials of 𝑋.

According to the Division theorem there exists polynomials 𝛼(𝑋),  𝛽(𝑋) of variable 𝑋 in ℜ[𝑋] such that 𝑑𝑃=𝛼(𝑋)(𝑋)𝑌𝑃().𝑑𝜉+𝛽𝑋,𝑌 (2.5)

We determine polynomials 𝛼(𝑋), 𝛽(𝑋), 𝑎_𝑖(𝑋) (𝑖=0,1,2,…) from (2.5) and, furthermore, obtain 𝑃(𝑋,𝑌).

Then substituting 𝑋=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 𝐸(𝑥,𝑡)=𝑒^{𝑖(𝑘𝑥+𝜔𝑡+𝜉0)}𝜑(𝑥,𝑡), (3.1) 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 𝑛_𝑡𝑡−𝑐²_𝑠𝑛_𝑥𝑥=𝛽𝜑²_𝑥𝑥,𝜑_𝑡+2𝛼𝑘𝜑_𝑥=0,𝛼𝜑_𝑥𝑥−𝛼𝑘²+𝜔𝜑=𝛿𝑛𝜑. (3.2)(3.3)(3.4) In the view of (3.3), we suppose 𝜑(𝑥,𝑡)=𝜑(𝜉)=𝜑𝑥−2𝛼𝑘𝑡+𝜉₁, (3.5) where 𝜉₁ is an arbitrary constant. Substituting (3.5) into (3.4), we infer that 𝑛(𝑥,𝑡)=𝛼𝜑_𝑥𝑥−𝜔+𝛼𝑘²𝜑=𝛿𝜑𝛼𝜑^′′(𝜉)(𝜉)−𝛿𝜑𝜔+𝛼𝑘². (3.6) Therefore, we can also assume 𝑛(𝑥,𝑡)=𝜓(𝜉)=𝜓𝑥−2𝛼𝑘𝑡+𝜉₁. (3.7) Substituting (3.7) into (3.2), and integrating the resultant equation twice with respect to 𝜉, we obtain 4𝛼²𝑘²−𝑐²_𝑠𝜓(𝜉)=𝛽𝜑²(𝜉)+𝐶^′𝜉+𝐶, (3.8) where 𝐶^′,  𝐶 are two arbitrary integration constants. For 𝑘=±𝑐_𝑠/2𝛼, (1.1) have one set of solution 𝐸₀()𝐶𝑥,𝑡=±^′𝑥±𝑐_𝑠𝑡+𝜉₀+𝐶−𝛽^1/2𝑒^{[𝑖(±(𝑐𝑠/2𝛼)𝑥+𝜔𝑡+𝜉0)]},𝑛₀()𝛼𝐶𝑥,𝑡=−^′24𝛿𝐶^′𝑥±𝑐_𝑠𝑡+𝜉₀+𝐶²−𝑐𝜔+^2𝑠.4𝛼 (3.9) For 𝑘≠±𝑐_𝑠/2𝛼, we put 𝐶^′=0 in (3.8) for the cause of technical. Thus (3.8) becomes 𝜓(𝜉)=𝛽4𝛼²𝑘²−𝑐²_𝑠𝜑²(𝜉)+𝐶. (3.10) Substituting (3.5), (3.7), and (3.10) into (3.4), we obtain 𝜑^′′(𝜉)−𝜔+𝛼𝑘²+𝐶𝛿𝛼𝜑(𝜉)−𝛿𝛽𝛼4𝛼²𝑘²−𝑐²_𝑠𝜑³(𝜉)=0. (3.11) Let 𝑙=(𝜔+𝛼𝑘²+𝐶𝛿)/𝛼,  𝑚=𝛿𝛽/𝛼(4𝛼²𝑘²−𝑐²_𝑠); thus (3.11) becomes the Liénard equation 𝜑^′′(𝜉)−𝑙𝜑(𝜉)−𝑚𝜑³(𝜉)=0. (3.12) Let 𝑋=𝜑(𝜉),  𝑌=𝑋^′, (3.12) can be converted to a system of nonlinear ODEs as follows: 𝑋^′𝑌=𝑌,_′=𝑙𝑋+𝑚𝑋³. (3.13) 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]𝑃(𝑋(𝜉),𝑌(𝜉))=^𝑚_𝑖=0𝑎_𝑖(𝑋)𝑌^𝑖=0, (3.14) where 𝑎_𝑖,  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 𝑑𝑃=𝑑𝜉𝜕𝑃𝜕𝑋𝜕𝑋+𝜕𝜉𝜕𝑃𝜕𝑌𝜕𝑌=𝛼(𝑋)(𝑋)𝑌𝜕𝜉+𝛽^𝑚_𝑖=0𝑎_𝑖(𝑋)𝑌^𝑖. (3.15)

Here we only consider the case of 𝑚=2.

Substituting (3.13) and (3.14) into (3.15), one gets 𝑑𝑃𝑑𝜉=𝑎^′₀(𝑋)𝑌+𝑎^′₁(𝑋)𝑌²+𝑎₁(𝑋)+2𝑎₂(𝑋)𝑌𝑌^′+𝑎^′₂(𝑥)𝑌³=𝛼(𝑋)+𝛽(𝑋)𝑌𝑎₀(𝑋)+𝑎₁(𝑋)𝑌+𝑎₂(𝑋)𝑌². (3.16) 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: 𝑎^′₂(𝑋)=𝛽(𝑋)𝑎₂(𝑋),𝑎^′₁(𝑋)=𝛽(𝑋)𝑎₁(𝑋)+𝛼(𝑋)𝑎₂(𝑋),𝑎^′₀(𝑋)=𝛽(𝑋)𝑎₀(𝑋)+𝛼(𝑋)𝑎₁(𝑋)−2𝑎₂(𝑋)𝑙𝑋+𝑚𝑋³,𝑎₁(𝑋)𝑙𝑋+𝑚𝑋³=𝛼(𝑋)𝑎₀(𝑋). (3.17)(3.18)(3.19)(3.20) Because 𝑎_𝑖(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 of 𝑎₁(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 𝑌²=𝑚2𝑋⁴+𝑙𝑋²+𝑑, (3.21) that is, 𝑋^′=±𝑚2𝑋⁴+𝑙𝑋²+𝑑. (3.22) Based on the discussion for different parameters, we can obtain the solutions of the nonlinear ordinary differential equation (3.22).

(1) For 𝑑=0, (3.22) admits the following five general solutions: 𝑋₁=±−2𝑙𝑚sec√𝑋−𝑙𝜉,𝑙<0,𝑚>0,₂=±−2𝑙𝑚csc√𝑋−𝑙𝜉,𝑙<0,𝑚>0,₃=±2𝑙𝑚csch√𝑋𝑙𝜉,𝑙𝑚>0,₄=±−2𝑙𝑚sech√𝑋𝑙𝜉,𝑙>0,𝑚<0,₅1=±𝑚/2,𝑙=0. (3.23) 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): 𝐸₁(𝑥,𝑡)=±2𝜔+𝛼𝑘²+𝐶𝛿4𝛼²𝑘²−𝑐²_𝑠𝛿𝛽sec−𝜔+𝛼𝑘²+𝐶𝛿𝛼𝜉𝑒^𝑖𝜂,𝑛₁()2𝑥,𝑡=−𝜔+𝛼𝑘²+𝐶𝛿𝛿sec²−𝜔+𝛼𝑘²+𝐶𝛿𝛼𝜉𝛼+𝐶,𝜔+𝛼𝑘²+𝐶𝛿<0,𝛼𝛽𝛿4𝛼²𝑘²−𝑐²_𝑠𝐸>0,₂(𝑥,𝑡)=±2𝜔+𝛼𝑘²+𝐶𝛿4𝛼²𝑘²−𝑐²_𝑠𝛿𝛽csc−𝜔+𝛼𝑘²+𝐶𝛿𝛼𝜉𝑒^𝑖𝜂,𝑛₂()2𝑥,𝑡=−𝜔+𝛼𝑘²+𝐶𝛿𝛿csc²−𝜔+𝛼𝑘²+𝐶𝛿𝛼𝜉𝛼+𝐶,𝜔+𝛼𝑘²+𝐶𝛿<0,𝛼𝛽𝛿4𝛼²𝑘²−𝑐²_𝑠𝐸>0,₃(𝑥,𝑡)=±2𝜔+𝛼𝑘²+𝐶𝛿4𝛼²𝑘²−𝑐²_𝑠𝛿𝛽csch𝜔+𝛼𝑘²+𝐶𝛿𝛼𝜉𝑒^𝑖𝜂,𝑛₃()=2𝑥,𝑡𝜔+𝛼𝑘²+𝐶𝛿𝛿csch²𝜔+𝛼𝑘²+𝐶𝛿𝛼𝜉𝛼+𝐶,𝜔+𝛼𝑘²+𝐶𝛿>0,𝛼𝛽𝛿4𝛼²𝑘²−𝑐²_𝑠𝐸>0,₄(𝑥,𝑡)=±2𝜔+𝛼𝑘²+𝐶𝛿4𝛼²𝑘²−𝑐²_𝑠𝛿𝛽sech𝜔+𝛼𝑘²+𝐶𝛿𝛼𝜉𝑒^𝑖𝜂,𝑛₄()2𝑥,𝑡=−𝜔+𝛼𝑘²+𝐶𝛿𝛿sech²𝜔+𝛼𝑘²+𝐶𝛿𝛼𝜉𝛼+𝐶,𝜔+𝛼𝑘²+𝐶𝛿>0,𝛼𝛽𝛿4𝛼²𝑘²−𝑐²_𝑠<0, (3.24)(3.25)(3.26)(3.27) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁,  𝜂=𝑘𝑥+𝜔𝑡+𝜂₀, 𝑘≠±𝑐_𝑠/2𝛼, 𝜔, 𝐶 are arbitrary parameters, and 𝜉₁, 𝜂₀ are two arbitrary constants. One has 𝐸₅(𝑥,𝑡)=±2𝛼4𝛼²𝑘²−𝑐²_𝑠1𝛿𝛽𝑥−2𝛼𝑘𝑡+𝜉₁𝑒^𝑖𝜂,𝑛₅(𝑥,𝑡)=2𝛼𝛿1𝑥−2𝛼𝑘𝑡+𝜉₁²−𝜔+𝛼𝑘²𝛿,𝛼𝛿𝛽4𝛼²𝑘²−𝑐²_𝑠>0, (3.28) where 𝜂=𝑘𝑥+𝜔𝑡+𝜂₀, 𝑘≠±𝑐_𝑠/2𝛼, 𝑘,  𝜔 are arbitrary parameters, and 𝜉₁,  𝜂₀ are two arbitrary constants.

Remark 3.1. In the above solutions, solutions 𝐸₁, 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) For 𝑑=𝑙²/2𝑚, we can obtain following four sets of explicit exact solutions to (3.22) 𝑋₆=±𝑙𝑚tan𝑙2𝜉+𝜉₂𝑋,𝑙>0,𝑚>0,₇=±𝑙𝑚cot𝑙2𝜉+𝜉₂𝑋,𝑙>0,𝑚>0,₈=±−𝑙𝑚tanh−𝑙2𝜉+𝜉₂𝑋,𝑙<0,𝑚>0,₉=±−𝑙𝑚coth−𝑙2𝜉+𝜉₂,𝑙<0,𝑚>0, (3.29) where 𝑙=(𝜔+𝛼𝑘²+𝐶𝛿)/𝛼,  𝑚=𝛿𝛽/𝛼(4𝛼²𝑘²−𝑐²_𝑠), 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): 𝐸₆(𝑥,𝑡)=±𝜔+𝛼𝑘²+𝐶𝛿4𝛼²𝑘²−𝑐²_𝑠𝛿𝛽tan𝜔+𝛼𝑘²+𝐶𝛿𝜉𝑒2𝛼^𝑖𝜂,𝑛₆(𝑥,𝑡)=𝜔+𝛼𝑘²+𝐶𝛿𝛿tan²𝜔+𝛼𝑘²+𝐶𝛿𝜉𝛼2𝛼+𝐶,𝜔+𝛼𝑘²+𝐶𝛿>0,𝛼𝛿𝛽4𝛼²𝑘²−𝑐²_𝑠𝐸>0,₇(𝑥,𝑡)=±𝜔+𝛼𝑘²+𝐶𝛿4𝛼²𝑘²−𝑐²_𝑠𝛿𝛽cot𝜔+𝛼𝑘²+𝐶𝛿𝜉𝑒2𝛼^𝑖𝜂,𝑛₇(𝑥,𝑡)=𝜔+𝛼𝑘²+𝐶𝛿𝛿cot²𝜔+𝛼𝑘²+𝐶𝛿𝜉𝛼2𝛼+𝐶,𝜔+𝛼𝑘²+𝐶𝛿>0,𝛼𝛿𝛽4𝛼²𝑘²−𝑐²_𝑠𝐸>0,₈(𝑥,𝑡)=±−𝜔+𝛼𝑘²+𝐶𝛿4𝛼²𝑘²−𝑐²_𝑠𝛿𝛽tanh−𝜔+𝛼𝑘²+𝐶𝛿𝜉𝑒2𝛼^𝑖𝜂,𝑛₈(𝑥,𝑡)=−𝜔+𝛼𝑘²+𝐶𝛿𝛿tanh²−𝜔+𝛼𝑘²+𝐶𝛿𝜉𝛼2𝛼+𝐶,𝜔+𝛼𝑘²+𝐶𝛿<0,𝛼𝛿𝛽4𝛼²𝑘²−𝑐²_𝑠𝐸>0,₉(𝑥,𝑡)=±−𝜔+𝛼𝑘²+𝐶𝛿4𝛼²𝑘²−𝑐²_𝑠𝛿𝛽coth−𝜔+𝛼𝑘²+𝐶𝛿𝜉𝑒2𝛼^𝑖𝜂,𝑛₉()2𝑥,𝑡=−𝜔+𝛼𝑘²+𝐶𝛿𝛿coth²−𝜔+𝛼𝑘²+𝐶𝛿𝜉𝛼2𝛼+𝐶,𝜔+𝛼𝑘²+𝐶𝛿<0,𝛼𝛿𝛽4𝛼²𝑘²−𝑐²_𝑠>0, (3.30)(3.31)(3.32)(3.33) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁,  𝜂=𝑘𝑥+𝜔𝑡+𝜂₀,  𝑘≠±𝑐_𝑠/2𝛼,  𝜔,  𝐶 are arbitrary parameters, and 𝜉₁, 𝜂₀ are two arbitrary constants.

Remark 3.2. The solutions 𝐸₆,  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) For 𝑑≠0, we obtain elliptic function solutions for (3.22) as follows: 𝑋₁₀=±−2(𝑙+1)𝑚sn𝑚𝑑22()𝜉,𝑑=−𝑙+1𝑚,𝑋₁₁()=±2𝑑1+𝑙𝑚(𝑙−1)^1/4cn𝑙+12𝑙𝜉,𝑑=²−1,𝑋2𝑚₁₂=±2𝑚dn𝑙+122()𝜉,𝑑=𝑙−1𝑚,𝑋₁₃=±2𝑚ns√2()−𝑙−1𝜉,𝑑=−𝑙+1𝑚,𝑋₁₄=±−𝑙−1𝑚^1/4nc𝑙+12𝑙𝜉,𝑑=²−1,𝑋2𝑚₁₅=±−2(𝑙−1)𝑚nd√2()2−𝑙𝜉,𝑑=𝑙−1𝑚,𝑋₁₆=±2(𝑙−1)𝑚sc√2()2−𝑙𝜉,𝑑=𝑙−1𝑚,𝑋₁₇=±𝑙²−12𝑚sd2+𝑙2𝑙𝜉,𝑑=²−1,𝑋2𝑚₁₈=±2𝑚cs√2()2−𝑙𝜉,𝑑=𝑙−1𝑚,𝑋₁₉=±−2(𝑙+1)𝑚cd√2()−𝑙−1𝜉,𝑑=−𝑙+1𝑚,𝑋₂₀=±2𝑚ds2+𝑙2𝑙𝜉,𝑑=²−1,𝑋2𝑚₂₁=±2𝑚dc√2()−𝑙−1𝜉,𝑑=−𝑙+1𝑚. (3.34)(3.35)(3.36)(3.37)(3.38)(3.39)(3.40)(3.41)(3.42)(3.43)(3.44)(3.45) 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) 𝐸₁₀(𝑥,𝑡)=±−2𝜔+𝛼𝑘²+𝐶𝛿+𝛼4𝛼²𝑘²−𝑐²_𝑠𝛿𝛽sn−𝜔+𝛼𝑘²+𝐶𝛿+𝛼𝛼𝜉𝑒^𝑖𝜂,𝑛₁₀()2𝑥,𝑡=−𝜔+𝛼𝑘²+𝐶𝛿+𝛼𝛿sn²−𝜔+𝛼𝑘²+𝐶𝛿+𝛼𝛼𝜉2+𝐶,𝑑=−𝜔+𝛼𝑘²+𝐶𝛿+𝛼4𝛼²𝑘²−𝑐²_𝑠,𝐸𝛿𝛽₁₁(𝑥,𝑡)=±𝜔+𝛼𝑘²+𝐶𝛿+𝛼4𝛼²𝑘²−𝑐²_𝑠𝛿𝛽cn𝜔+𝛼𝑘²+𝐶𝛿+𝛼𝜉𝑒2𝛼^𝑖𝜂,𝑛₁₁(𝑥,𝑡)=𝜔+𝛼𝑘²+𝐶𝛿+𝛼𝛿cn²𝜔+𝛼𝑘²+𝐶𝛿+𝛼𝜉2𝛼+𝐶,𝑑=𝜔+𝛼𝑘²+𝐶𝛿²−𝛼²4𝛼²𝑘²−𝑐²_𝑠,𝐸2𝛼𝛿𝛽₁₂(𝑥,𝑡)=±2𝛼4𝛼²𝑘²−𝑐²_𝑠𝛿𝛽dn2𝛼−𝜔+𝛼𝑘²+𝐶𝛿𝛼𝜉𝑒^𝑖𝜂,𝑛₁₂(𝑥,𝑡)=2𝛼𝛿dn²2𝛼−𝜔+𝛼𝑘²+𝐶𝛿𝛼𝜉2+𝐶,𝑑=𝜔+𝛼𝑘²+𝐶𝛿−𝛼4𝛼²𝑘²−𝑐²_𝑠,𝐸𝛿𝛽₁₃(𝑥,𝑡)=±2𝛼4𝛼²𝑘²−𝑐²_𝑠𝛿𝛽ns−𝜔+𝛼𝑘²+𝐶𝛿+𝛼𝛼𝜉𝑒^𝑖𝜂,𝑛₁₃(𝑥,𝑡)=2𝛼𝛿ns²−𝜔+𝛼𝑘²+𝐶𝛿+𝛼𝛼𝜉2+𝐶,𝑑=−𝜔+𝛼𝑘²+𝐶𝛿+𝛼4𝛼²𝑘²−𝑐²_𝑠,𝐸𝛿𝛽₁₄(𝑥,𝑡)=±−𝜔+𝛼𝑘²+𝐶𝛿−𝛼4𝛼²𝑘²−𝑐²_𝑠𝛿𝛽nc𝜔+𝛼𝑘²+𝐶𝛿+2𝛼𝜉𝑒2𝛼^𝑖𝜂,𝑛₁₄(𝑥,𝑡)=−𝜔+𝛼𝑘²+𝐶𝛿−𝛼𝛿nc²𝜔+𝛼𝑘²+𝐶𝛿+2𝛼𝜉2𝛼+𝐶,𝑑=𝜔+𝛼𝑘²+𝐶𝛿²−𝛼²4𝛼²𝑘²−𝑐²_𝑠,𝐸2𝛼𝛿𝛽₁₅(𝑥,𝑡)=±−2𝜔+𝛼𝑘²+𝐶𝛿−𝛼4𝛼²𝑘²−𝑐²_𝑠𝛿𝛽nd2𝛼−𝜔+𝛼𝑘²+𝐶𝛿𝜉𝑒2𝛼^𝑖𝜂,𝑛₁₅()2𝑥,𝑡=−𝜔+𝛼𝑘²+𝐶𝛿−𝛼𝛿nd²2𝛼−𝜔+𝛼𝑘²+𝐶𝛿𝜉22𝛼+𝐶,𝑑=𝜔+𝛼𝑘²+𝐶𝛿−𝛼4𝛼²𝑘²−𝑐²_𝑠,𝐸𝛿𝛽₁₆(𝑥,𝑡)=±2𝜔+𝛼𝑘²+𝐶𝛿−𝛼4𝛼²𝑘²−𝑐²_𝑠𝛿𝛽sc2𝛼−𝜔+𝛼𝑘²+𝐶𝛿𝜉𝑒2𝛼^𝑖𝜂,𝑛₁₆()=2𝑥,𝑡𝜔+𝛼𝑘²+𝐶𝛿−𝛼𝛿sc²2𝛼−𝜔+𝛼𝑘²+𝐶𝛿𝜉22𝛼+𝐶,𝑑=𝜔+𝛼𝑘²+𝐶𝛿−𝛼4𝛼²𝑘²−𝑐²_𝑠,𝐸𝛿𝛽₁₇(𝑥,𝑡)=±𝜔+𝛼𝑘²+𝐶𝛿²−𝛼²4𝛼²𝑘²−𝑐²_𝑠2𝛼𝛿𝛽sd𝜔+𝛼𝑘²+𝐶𝛿+2𝛼𝜉𝑒2𝛼^𝑖𝜂,𝑛₁₇(𝑥,𝑡)=𝜔+𝛼𝑘²+𝐶𝛿²−𝛼²2𝛼𝛿sd²𝜔+𝛼𝑘²+𝐶𝛿+2𝛼𝜉2𝛼+𝐶,𝑑=𝜔+𝛼𝑘²+𝐶𝛿²−𝛼²4𝛼²𝑘²−𝑐²_𝑠,𝐸2𝛼𝛿𝛽₁₈(𝑥,𝑡)=±2𝛼4𝛼²𝑘²−𝑐²_𝑠𝛿𝛽cs2𝛼−𝜔+𝛼𝑘²+𝐶𝛿𝜉𝑒2𝛼^𝑖𝜂,𝑛₁₈(𝑥,𝑡)=2𝛼𝛿cs²2𝛼−𝜔+𝛼𝑘²+𝐶𝛿𝜉22𝛼+𝐶,𝑑=𝜔+𝛼𝑘²+𝐶𝛿−𝛼4𝛼²𝑘²−𝑐²_𝑠,𝐸𝛿𝛽₁₉(𝑥,𝑡)=±−2𝜔+𝛼𝑘²+𝐶𝛿+𝛼4𝛼²𝑘²−𝑐²_𝑠𝛿𝛽cd−𝜔+𝛼𝑘²+𝐶𝛿+𝛼𝛼𝜉𝑒^𝑖𝜂,𝑛₁₉()2𝑥,𝑡=−𝜔+𝛼𝑘²+𝐶𝛿+𝛼𝛿cd²−𝜔+𝛼𝑘²+𝐶𝛿+𝛼𝛼𝜉2+𝐶,𝑑=−𝜔+𝛼𝑘²+𝐶𝛿+𝛼4𝛼²𝑘²−𝑐²_𝑠,𝐸𝛿𝛽₂₀(𝑥,𝑡)=±2𝛼4𝛼²𝑘²−𝑐²_𝑠𝛿𝛽ds𝜔+𝛼𝑘²+𝐶𝛿+2𝛼𝜉𝑒2𝛼^𝑖𝜂,𝑛₂₀(𝑥,𝑡)=2𝛼𝛿ds²𝜔+𝛼𝑘²+𝐶𝛿+2𝛼𝜉2𝛼+𝐶,𝑑=𝜔+𝛼𝑘²+𝐶𝛿²−𝛼²4𝛼²𝑘²−𝑐²_𝑠,𝐸2𝛼𝛿𝛽₂₁(𝑥,𝑡)=±2𝛼4𝛼²𝑘²−𝑐²_𝑠𝛿𝛽dc−𝜔+𝛼𝑘²+𝐶𝛿+𝛼𝛼𝜉𝑒^𝑖𝜂,𝑛₂₁(𝑥,𝑡)=2𝛼𝛿dc²−𝜔+𝛼𝑘²+𝐶𝛿+𝛼𝛼𝜉2+𝐶,𝑑=−𝜔+𝛼𝑘²+𝐶𝛿+𝛼4𝛼²𝑘²−𝑐²_𝑠,𝛿𝛽 (3.46)(3.47)(3.48)(3.49)(3.50)(3.51)(3.52)(3.53)(3.54)(3.55)(3.56)(3.57) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁,  𝜂=𝑘𝑥+𝜔𝑡+𝜂₀,  𝑘≠±𝑐_𝑠/2𝛼,  𝜔,  𝐶 are arbitrary parameters, and 𝜉₁,  𝜂₀ are two arbitrary constants.

(4) For 𝑙=0, 𝑚≠0, from (3.22) we have 𝑋^′=±𝑚2𝑋⁴+𝑑. (3.58) Let 𝑋²=𝑍; (3.58) becomes 𝑍^′(𝜉)=±2𝑚𝑍³+4𝑑𝑍. (3.59) While 𝑚>0, the above equation possesses a Weierstrass elliptic function doubly periodic wave type solution 𝑍=℘𝑚2𝜉,−8𝑑𝑚,0. (3.60) So (3.58) admits a Weierstrass elliptic function doubly periodic wave type solution 𝑋₂₂=±℘𝑚2𝜉,−8𝑑𝑚,0. (3.61) 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 𝐸₂₂(𝑥,𝑡)=±℘𝛿𝛽2𝛼4𝛼²𝑘²−𝑐²_𝑠𝜉,−8𝑑𝛿𝛽𝛼4𝛼²𝑘²−𝑐²_𝑠,0𝑒^{𝑖(𝑘𝑥+𝜔𝑡+𝜉0)},𝑛₂₂()=𝛽𝑥,𝑡4𝛼²𝑘²−𝑐²_𝑠℘𝛿𝛽2𝛼4𝛼²𝑘²−𝑐²_𝑠𝜉,−8𝑑𝛿𝛽𝛼4𝛼²𝑘²−𝑐²_𝑠,0+𝐶, (3.62) where 𝜉=𝑥−2𝛼𝑘𝑡+𝜉₁.

Remark 3.3. The above twelve explicit exact Jacobi elliptic doubly periodic wave solutions 𝐸₁₀,  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 of deg[𝑎₁(𝑋)]−1=deg[𝛼(𝑋)].

In this case, we assume that deg[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 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. (3.63) By analyzing all kinds of possibilities, we have the following.(1)While 𝐵₀=𝐶₀=0, it leads to a contradiction.(2)While 𝐵₀≠0,𝐶₀=0, it also leads to a contradiction.(3)While 𝐵₀=0,𝐶₀≠0, we can derive that 𝐴₀𝑙√=±2𝑚𝑚,𝐴₁=0,𝐴₂=±√2𝑚,𝐶₀=𝑙²2𝑚,𝐶₁=𝐶₃=0,𝐶₂𝐶=𝑙,₄=𝑚2. (3.64)Setting (3.64) in (3.14) yields 𝑋^′2±√2𝑚𝑋²+𝑙𝑚√2𝑚𝑋^′+𝑚2𝑋⁴+𝑙𝑋²+𝑙²2𝑚=0, (3.65) that is, 𝑑𝑋𝑋²+𝑙/𝑚=±𝑚2𝜉+𝜉₀. (3.66) 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 for 𝑛 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.

Acknowledgments

This work is supported by the NSF of China (40890150, 40890153, 11271090), the Science and Technology Program (2008B080701042) of Guangdong Province, and Natural Science foundation of Guangdong Province (S2012010010121). This work is also supported by the Visiting Scholar Program of Chern Institute of Mathematics at Nankai University when the authors worked as visiting scholars. The authors would like to express their hearty thanks to Chern Institute of Mathematics which provided very comfortable research environments to them. The authors would like to thank Professor Wang Mingliang for his helpful suggestions.