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**The first-integral method and abundant explicit exact solutions to the Zakharov equations.**
*(English)*
Zbl 1308.65179

Summary: 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.

### MSC:

65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |

35Q53 | KdV equations (Korteweg-de Vries equations) |

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\textit{Y. Shang} and \textit{X. Zheng}, J. Appl. Math. 2012, Article ID 818345, 16 p. (2012; Zbl 1308.65179)

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### References:

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