# zbMATH — the first resource for mathematics

The extended hyperbolic functions method and new exact solutions to the Zakharov equations. (English) Zbl 1143.65083
Summary: The multiple exact solutions for the nonlinear evolution equations describing the interaction of laser-plasma are developed. The extended hyperbolic function method are employed to reveal these new solutions. The solutions 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 solitary wave solutions of a compound of the bell-type and the kink-type for $$n$$ and $$E$$, the singular traveling wave solutions, periodic traveling wave solutions of triangle function types, and solitary wave solutions of rational function types. In addition to re-deriving all known solutions in a systematic way, several new and more general solutions can be obtained by using our method.

##### MSC:
 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 35Q53 KdV equations (Korteweg-de Vries equations)
RAEEM
Full Text:
##### References:
 [1] Zakharov, V.E., Collapse of Langmuir waves, Sov. phys. JETP, 35, 908-914, (1972) [2] Zakharov, V.E.; Syankh, V.S., The nature of the self-focusing singularity, Sov. phys. JETP, 41, 465-468, (1976) [3] Thornhill, S.G.; Haar, Dter, Langmuir turbulence and modulation instability, Phys lett. sect. C phys. rep., 43, 43-99, (1978) [4] Makankov, V.G., Dynamics of classical solitons (in nonintegrable systems), Phys letter. sect. C phys. rep., 35, 1, 1-128, (1978) [5] Chen, H.H.; Liu, C.S., Phys. fluids, 21, 3, (1978) [6] Goldman, M.V.; Nicholson, D.R., Phys. rev. lett., 41, 406, (1978) [7] Kono, M.; Skoric, M.M.; Ter Har, D., J. plasma phys., 26, 123, (1981) [8] Zhang, Wei-Guo; Chang, Qian-Shun; Fan, En-Gui, Methods of judging shape of solitary wave and solution formulae for some evolution equations with nonlinear terms of high order, J. math. anal. appl., 287, 1-18, (2003) · Zbl 1040.35106 [9] Zhao, Chang-Hai; Sheng, Zheng-Mao, Explicit traveling wave solutions for Zakharov equations, Acta phys. sin., 53, 6, 1629-1634, (2004) · Zbl 1202.35287 [10] Huang, Ding-Jiang; Zhang, Hong-Qing, Extended hyperbolic function method and new exact solitary wave solutions of Zakharov equations, Acta phys. sin., 53, 8, 2434-2438, (2004) · Zbl 1202.81039 [11] Zhang, Jin-Liang; Wang, Ming-Liang, Complex tanh-function expansion method and exact solutions to two systems of nonlinear wave equations, Commun. theor. phys, 42, 4, 491, (2004) · Zbl 1167.35497 [12] Liu, Shi-Da; Fu, Zun-Tao; Liu, Shi-Kuo; Zhao, Qiang, Acta. phys. sin., 51, 4, 718, (2003) [13] Zhang, Jin-Liang; wang, Yue-Ming; Wang, Ming-Liang; Fang, Zong-de, Exact solutions for two classes of nonlinear wave equations, Acta. phys. sin., 52, 7, 1574, (2003) · Zbl 1202.35285 [14] Wang, Ming-Liang; Wang, Yue-Ming; Zhang, Jin-Liang, The periodic wave solutions for two systems of nonlinear wave equations, Chin. phys., 12, 12, 1341-1348, (2003) [15] Shang, Ya-dong, Explicit and exact solutions for a generalized long – short wave equations with strong nonlinear term, Chaos solitons fract., 26, 527-539, (2005) · Zbl 1070.35039 [16] Conte, R.; Musette, M., Link between solitary waves and projective Riccati equations, J. phys. A: math. gen., 25, 5609-5612, (1992) · Zbl 0782.35065 [17] Zhang, Gui-Xu; Li, Zhi-Bin; Duan, Yi-Shi, Exact solitary wave solutions of nonlinear exact solitary wave solutions of nonlinear wave equations, Sci. China, 44, 3, 396-401, (2001) · Zbl 1054.35032 [18] Li, Zhi-Bin; Liu, Yin-Ping, RAEEM: a Maple package for finding a series of exact traveling wave solutions for nonlinear evolution equations, Comput. phys. commun., 163, 191-201, (2004) · Zbl 1196.35009 [19] Yan, Zhen-Ya, Generalized method and its application in the higher-order nonlinear schrodinger equation in nonlinear optical fibres, Chaos solitons fract., 16, 759-766, (2003) · Zbl 1035.78006 [20] Chen, Yanze; Ding, Xinwei, Exact traveling wave solutions of nonlinear evolution equations in $$(1 + 1)$$ and $$(2 + 1)$$ dimensions, Nonlinear anal., 61, 1005-1013, (2005) · Zbl 1086.34501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.