Solutions of sine Gordon equation by generalized exponential function methods
Keywords:
Sine Gordon equations, exp-function methods, multiwave solutionsAbstract
The sine Gordon equation (sG) is hyperbolic partial differential equation involving the d’Alembert operator and the sine of the unknown function. The importance of the equation grew from 1970, when led to kink and antikink solitons discovery. In the development of soliton theory, the multiwave solutions have gradually become a field of study of nonlinear science. Such multiwave solutions can be obtained by the exp function method proposed by He and Wu in 2006, the method is used in solving different classes of nonlinear differential equations such as KdV, mKdV and sGs. In this paper we describe the exp-function method in the solution of the sG equation, the results presented are for soliton solutions for single, two and three wave. We chose the positive sign in the solution and found that for negative values Z the amplitude of the solution is close to zero, while for positive values Z it is close to 2pi.
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P.G. Drazin and R. S. Johnson. Solitons: an Introduction. Cambridge University Press, Cambridge, 1996.
N. J. Zabusky and M. D. Kruskal. Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett., 15, 240-243, 1965.
M. Toda, Nonlinear waves and solitons. Springer, Berlin, 1989.
Y. Kivshar and G. Agrawal, Optical solitons, Academic Press, USA, 2003.
M. Segev, Solitons: A Universal Phenomenon of Self-Trapped Wave Packets, Opt. Photonics news, 13, 27-29, 2002.
A. Scott, Nonlinear science. Emergence and dynamics of coherent structures, Oxford University Press, 1999.
M. A. Matias y J. Guemez, Stabilization of chaos by proportional pulses in the sistema variables, Phys. Rev. Lett. 72, 1455-1462, 1994.
E. Trias, J. J. Mazo and T.P. Orlando, Discrete breathers in nonlinear lattices: Experimental detection in a Josephson-junction array, Phys. Rev. Lett. 84, 741-744, 2000.
P. Binder, Observation of breathers in Joshepson Ladders, Phys. Rev. Lett. 84, 745, 2000.
C. Pooles, H. Farach and R. Creswick, Superconductivity, Academic Press, Columbia, 1995.
W. J. Pierson, M. A. Donelan and W. H. Hui, Linear and nonlinear propagation of water wave groups, J. Geophys. Oceans, 97, 5607-5621, 1992.
C. S. Gardner, J. M. Green, M. D. Kruskal and R. Miura, Methods for solving the Korteweg de Vries equation, Phys. Rev. Lett. 19, 1095-1097, 1967.
S. P. Burstev , V. E. Zakharov and A. V. Mikhailov, The inverse scattering methods with variable spectral parameter, Theor. Math. Phy. 70, 232-241, 1987.
R. K. Dodd and R. K. Bullough, Polinomial conserved densities for the sine Gordon equation, Proc. R. Soc. Lond. A 352, 481-503, 1977.
J. He and L. Zhang, Generalized solitary solution and compacton-like solution of the Jaulent–Miodek equations using the Exp-function method, Phys. Lett. A 372, 1044, 2008.
S. Zhang, J. Wang, A. Peng and B. Cai, A generalized exp-function method for multiwave solutions of sine Gordon equation, J. Phys. Pramana, 81, 763-773, 2013.
S. Zhang, W. Wang and J. Tong, The Exp-Function Method for the Riccati Equation and Exact Solutions of Dispersive Long Wave Equations, Z. Naturfosch, 63, 663 – 670, 2008.
A. Ebaid, Application of the exp function method for solving some evolution equations with nonlinear terms any orders, Z. Naturfosch, 65, 1039-1044, 2010.
J. Saletan, Classical dynamics a contemporary approach, Cambridge University Press, Cambridge, 1998.
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