^{1, 2}

^{1, 2}

^{1}

^{2}

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

Zakharov equations

More recently, some authors considered the exact and explicit solutions of the system of Zakharov equations by different methods in [

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 (

The rest of this paper is organized as follows. In Section

The first-integral method, which is based on the ring theory of commutative algebra, was first proposed by Prof. Feng Zhaosheng [

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 [

The main steps of this method are summarized as follows.

Given a system of nonlinear partial differential equations, for example, in two independent variables

According to the Division theorem there exists polynomials

We determine polynomials

Then substituting

In this section we will employ the first integral method to construct abundant explicit exact traveling wave solutions to (

In order to transfer (

Here we only consider the case of

Substituting (

(a) In the case of

In this case, (

(1) For

In the above solutions, solutions

(2) For

Combining (

The solutions

(3) For

(4) For

The above twelve explicit exact Jacobi elliptic doubly periodic wave solutions

(b) In the case of

In this case, we assume that

While

While

While

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

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.