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

Shang, Yadong; Zheng, Xiaoxiao

J. Appl. Math. 2012, Article ID 818345, 16 p. (2012).

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)

References:

[1]	V. E. Zakharov, “Collapse of Langmuir waves,” Soviet Journal of Experimental and Theoretical Physics, vol. 35, pp. 908-914, 1972.
[2]	V. E. Zakharov and V.S. Syankh, “The nature of the self-focusing singularity,” Soviet Journal of Experimental and Theoretical Physics, vol. 35, pp. 908-914, 1972.
[3]	J. A. Pava and C. B. Brango, “Orbital stability for the periodic Zakharov system,” Nonlinearity, vol. 24, no. 10, pp. 2913-2932, 2011. · Zbl 1270.35078
[4]	Y. Shang, Y. Huang, and W. Yuan, “The extended hyperbolic functions method and new exact solutions to the Zakharov equations,” Applied Mathematics and Computation, vol. 200, no. 1, pp. 110-122, 2008. · Zbl 1143.65083
[5]	D. Huang and H. Zhang, “Extended hyperbolic function method and new exact solitary wave solutions of Zakharov equations,” Acta Physica Sinica, vol. 53, no. 8, pp. 2434-2438, 2004. · Zbl 1202.81039
[6]	S. Liu, Z. Fu, S. Liu, and Q. Zhao, “The envelope periodic solutions to nonlinear wave equations with Jacobi elliptic function,” Acta Physica Sinica, vol. 51, no. 4, pp. 718-722, 2002. · Zbl 1202.35303
[7]	G. Wu, M. Zhang, L. Shi, W. Zhang, and J. Han, “An extended expansion method for Jacobi elliptic functions and new exact periodic solutions of Zakharov equations,” Acta Physica Sinica, vol. 56, no. 9, pp. 5054-5059, 2007. · Zbl 1150.35323
[8]	C. Zhao and Z. Sheng, “Explicit travelling wave solutions for Zakharov equations,” Acta Physica Sinica, vol. 53, no. 6, pp. 1629-1634, 2004. · Zbl 1202.35287
[9]	Z. Feng, “The first integer method to study the Burgers-Korteweg-de Vries equation,” Journal of Physics A, vol. 35, no. 2, pp. 343-349, 2002. · Zbl 1040.35096
[10]	Z. Feng, “On explicit exact solutions to the compound Burgers-KdV equation,” Physics Letters A, vol. 293, no. 1-2, pp. 57-66, 2002. · Zbl 0984.35138
[11]	Z. Feng, “Exact solution to an approximate sine-Gordon equation in (n+1)-dimensional space,” Physics Letters A, vol. 302, no. 2-3, pp. 64-76, 2002. · Zbl 0998.35046
[12]	Z. Feng, “The first-integral method to the Two-dimensional Burgers-Korteweg-de Vries equationn,” Physics Letters A, vol. 302, pp. 57-66., 2002.
[13]	Z. Feng and R. Knobel, “Traveling waves to a Burgers-Korteweg-de Vries-type equation with higher-order nonlinearities,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1435-1450, 2007. · Zbl 1119.35075
[14]	Z. Feng and G. Chen, “Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth,” Discrete and Continuous Dynamical Systems A, vol. 24, no. 3, pp. 763-780, 2009. · Zbl 1360.34100
[15]	K. Hosseini, R. Ansari, and P. Gholamin, “Exact solutions of some nonlinear systems of partial differential equations by using the first integral method,” Journal of Mathematical Analysis and Applications, vol. 387, no. 2, pp. 807-814, 2012. · Zbl 1254.35044
[16]	B. Hong, D. Lu, and F. Sun, “The extended Jacobi elliptic functions expansion method and new exact solutions for the Zakharov equations,” Word Journal of Modelling and Simulation, vol. 5, no. 3, pp. 216-224, 2009.