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 fornandE, the solitary wave solutions of kink-type forEand bell-type forn, 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.
Highlights
Zakharov equations 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 laserplasma
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
Summary
Zakharov equations 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 laserplasma. Some authors considered the exact and explicit solutions of the system of Zakharov equations by different methods in 4–8. 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. 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 gives abundant explicit exact solitary wave solutions, periodic wave solutions of triangle function, and provides Jacobi elliptic function and Weierstrass elliptic function doubly periodic wave solutions.
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