Non-Traveling Wave Solutions for the (2+1)-Dimensional Breaking Soliton System

Yuanming Chen Chen, Songhua Ma

College of Sciences, Zhejiang Lishui University, Lishui 323000, China.

DOI: 10.4236/am.2012.38122                

Abstract

In this work, starting from the (G'/G)-expansion method and a variable separation method, a new non-traveling wave general solutions of the (2+1)-dimensional breaking soliton system are derived. By selecting appropriately the arbitrary functions in the solutions, special soliton-structure excitations and evolutions are studied.

Keywords

(G'/G)-Expansion Method; Variable Separation Approach; Breaking Soliton System, Non-Traveling Wave Solution

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	S. Y. Lou and X. Y. Tang, “Fractal Solutions of the Nizhnik-Novikov-Veselov Equation,” Chinese Physica Letter, Vol. 19, No. 6, 2002, pp. 769-771. doi:10.1088/0256-307X/19/6/308
[2]	S. Y. Lou and X. B. Hu, “Infinitely Many Lax Pairs and Symmetry Constraints of the KP Equation,” Journal of Mathematical Physics, Vol. 38, No. 6, 1997, Article ID: 6401. doi:10.1063/1.532219
[3]	X. Y. Tang and S. Y. Lou, “Localized Excitations in (2+ 1)-Dimensional Systems,” Physical Review E, Vol. 66, No. 4, 2002, Article ID: 046601. doi:10.1103/PhysRevE.66.046601
[4]	S. Wang, X. Y. Tang and S. Y. Lou, “Soliton Fission and Fusion: Burgers Equation and Sharma-Tasso-Olver Equation,” Chaos, Solitons & Fractals, Vol. 19, No. 1, 2004, pp. 231-239. doi:10.1016/j.chaos.2003.10.014
[5]	C. L. Zheng, “Localized Coherent Structures with Chaotic and Fractal Behaviors in a (2+1)-Dimensional Modified Dispersive Water-Wave System,” Communications in Theoretial Physics, Vol. 40, No. 2, 2003, pp. 25-32.
[6]	C. L. Zheng and J. M. Zhu, “Fractal Dromion, Fractal Lump, and Multiple Peakon Excitations in a New (2+1)-Dimensional Long Dispersive Wave System,” Communications in Theoretial Physics, Vol. 39, No. 1, 2003, pp. 261-266.
[7]	C. L. Zheng and J. F. Zhang, “Folded Localized Excitations in a Generalized (2+l)-Dimensional Perturbed Nonlinear Schrodinger System,” Communications in Theoretial Physics, Vol. 40, No. 2, 2003, p. 385.
[8]	P. A. Clarkson and M. D. Kruskal, “New Similarity Reductions of the Boussinesq Equation,” Journal of Mathematical Physics, Vol. 30, No. 10, 1989, Article ID: 2201. doi:10.1063/1.528613
[9]	S. Y. Lou, “A Note on the New Similarity Reductions of the Boussinesq Equation,” Physica Letter A, Vol. 151, No. 3-4, 1990, pp. 133-135. doi:10.1016/0375-9601(90)90178-Q
[10]	X. Y. Tang, J. Lin and S. Y. Lou, “Conditional Similarity Solutions of the Boussinesq Equation,” Communications in Theoretial Physics, Vol. 35, No. 3, 2001, pp. 399-404.
[11]	S. Y. Lou and X. Y. Tang, “Conditional Similarity Reduction Approach: Jimbo-Miwa equation,” Chinese Physics, Vol.10, No. 10, 2001, p. 897. doi:10.1088/1009-1963/10/10/303
[12]	S. Y. Lou, X. Y. Tang and J. Lin, “Similarity and Conditional Similarity Reductions of a (2+1)-Dimensional KdV Equation via a Direct Method,” Journal Mathematical Physics, Vol. 41, No. 12, 2000, Article ID: 8286. doi:10.1063/1.1320859
[13]	S. H. Ma and J. P. Fang, “New Exact Solutions for the Related Schr?dinger Equation and the Temporal-soliton and Soliton-Impulse,” Acta Physica Sinica, Vol. 55, No. 11, 2006, pp. 5611-5616.
[14]	S. H. Ma, X. H. Wu, J. P. Fang and C. L. Zheng, “Chaotic Solitons for the (2+1)-Dimensional Modified Dispersive Water-Wave System,” Zeitschrift für Nnturforschung A, Vol. 61, No. 1, 2006, pp. 249-252.
[15]	S. H. Ma, J. P. Fang and C. L. Zheng, “Folded Locailzed Excitations and Chaotic Patterns in a (2+1)-Dimensional Soliton System,” Zeitschrift für Nnturforschung A, Vol. 62, No. 1, 2008, pp. 121-126.
[16]	S. H. Ma, J. Y. Qiang and J. P. Fang, “The Interaction between Solitons and Chaotic Behaviours of (2+1)-Dimensional Boiti-Leon-Pempinelli System,” Acta Physics Sinica, Vol. 56, No. 2, 2007, pp. 620-626.
[17]	S. H. Ma, J. P. Fang and H. P. Zhu, “Dromion Soliton Waves and the Their Evolution in the Background of Jacobi Sine Waves,” Acta Physics Sinica, Vol. 56, No. 8, 2007, pp. 4319-4325.
[18]	S. L. Zhang, X. N. Zhu, Y. M. Wang and S. Y. Lou, “Extension of Variable Separable Solutions for Nonlinear Evolution Equations,” Communications in Theoretial Physics, Vol. 49, No. 1, 2008, pp. 829-832.
[19]	S. L. Zhang and S. Y. Lou, “Functional Variable Separation for Extended Nonlinear Elliptic Equations,” Communications in Theoretial Physics, Vol. 48, No. 3, 2007, pp. 385-390. doi:10.1088/0253-6102/48/3/001
[20]	D. J. Zhang, “Singular Solutions in Casoratian Form for Two Differential-Difference Equations,” Chaos, Solitons and Fractals, Vol. 23, No. 4, 2005, pp. 1333-1350.
[21]	D. J. Zhang, “The N-Soliton Solutions of Some Soliton Equations with Self-Consistent Sources,” Chaos, Solitons and Fractals, Vol. 18, No. 1, 2003, pp. 31-43. doi:10.1016/S0960-0779(02)00636-7
[22]	C. L. Zheng and L. Q. Chen, “Solitons with Fission and Fusion Behaviors in a Variable Coefficient BroerCKaup System,” Chaos, Solitons and Fractals, Vol. 24, No. 5, 2005, pp. 1347-1351. doi:10.1016/j.chaos.2004.09.069
[23]	C. L. Zheng and J. P. Fang, “New Exact Solutions and Fractal Patterns of Generalized BroerCKaup System via a Mapping Approach,” Chaos, Solitons and Fractals, Vol. 27, No. 5, 2006, pp. 1321-1327. doi:10.1016/j.chaos.2005.04.114
[24]	M. L. Wang, X. Z. Li and J. L. Zhang, “The (G'/G)-Expansion Method and Travelling Wave Solutions of Nonlinear Evolution Equations in Mathematical Physics,” Physics Letter A, Vol. 372, No. 4, 2008, pp. 417-423. doi:10.1016/j.physleta.2007.07.051
[25]	S. Zhang and J. L. Tong, “A Generalized (G'/G)-Expansion Method for the mKdV Equation with Variable Coefficients,” Physics Letter A, Vol. 372, No. 13, 2008, pp. 2254-2257. doi:10.1016/j.physleta.2007.11.026
[26]	A. Bekir, “Application of the (G'/G)-Expansion Method for Nonlinear Evolution Equations,” Physics Letter A, Vol. 372, No. 19, 2008, pp. 3400-3406. doi:10.1016/j.physleta.2008.01.057
[27]	J. Zhang, X. L. Wei and Y. J. Lu, “A Generalized (G'/G)-Expansion Method and Its Applications,” Physics Letter A, Vol. 372, No. 20, 2008, pp. 3653-3658. doi:10.1016/j.physleta.2008.02.027
[28]	E. M. E. Zayed and K. A. Gepreel, “The (G/G)-Expansion Method for Finding Traveling Wave Solutions of Nonlinear Partial Differential Equations in Mathematical Physics,” Journal of Mathematical Physics, Vol. 50, No. 1, 2009, Article ID: 013502. doi:10.1063/1.3033750
[29]	B. Q. Li, Y. L. Ma, C. Wang, M. P. Xu and Y. Li, “Folded Soliton with Periodic Vibration for a Nonlinear Coupled Schr?dinger System,” Acta Physica Sinica, Vol. 60, No. 6, 2011, Article ID: 060203.
[30]	B. Q. Li, Y. L. Ma and M. P. Xu, “(G'/G)-Expansion Method and Novel Fractal Structures for High-Dimensional Nonlinear Physical Equation,” Acta Physica Sinica, Vol. 59, No. 3, 2010, pp. 1409-1415.
[31]	R. Radha and M. Laskshmanan, “Dromion Like Structures in the (2+1)-Dimensional Breaking Soliton Equation,” Physics Letter A, Vol. 197, No. 1, 1995, pp. 7-12. doi:10.1016/0375-9601(94)00926-G
[32]	S. H. Ma, J. Y. Qiang and J. P. Fang, “Annihilation Solitons and Chaotic Solitons for the (2+1)-Dimensional Breaking Soliton System,” Communications in Theoretial Physics, Vol. 48, No. 2, 2007, p. 662.