Traveling Wave Solutions and Kind Wave Excitations for the (2 + 1)-Dimensional Dissipative Zabolotskaya-Khokhlov Equation

Xiajie Liu, Chunliang Mei, Songhua Ma

College of Science, Lishui University, Lishui, China.

DOI: 10.4236/am.2013.412215                

Abstract

In this work, with the help of the symbolic computation system Maple and the Riccati mapping approach and a linear variable separation approach, a new family of traveling wave solutions of the (2 + 1)-dimensional dissipative Zabolotskaya-Khokhlov equation (DZK) is derived. Based on the derived solitary wave solution, some novel kind wave excitations are investigated.

Keywords

Mapping Approach; Dissipative Zabolotskaya-Khokhlov Equation; Traveling Wave Solution; Kind Wave Excitation

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	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.
[2]	M. M. Helal, M. L. Mekky and E. A. Mohamed, “The Characteristic Function Method and Its Application to (1 + 1)-Dimensional Dispersive Long Wave Equation,” Applied Mathematics, Vol. 3, No. 1, 2012, pp. 12-18.
[3]	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. 12, 1997, p. 6401. http://dx.doi.org/10.1063/1.532219
[4]	S. Wang, X. Y. Tang and S. Y. Lou, “Soliton Fission and Fusion: Burgers Equation and Sharma-Tasso-Olver Equation,” Chaos, Solitons and Fractals, Vol. 19, No. 1, 2004, pp.231-239. http://dx.doi.org/10.1016/j.chaos.2003.10.014
[5]	P. A. Clarkson and M. D. Kruskal, “New Similarity Reductions of the Boussinesq Equation,” Journal of Mathematical Physics, Vol. 30, No. 10, 1989, p. 2201. http://dx.doi.org/10.1063/1.528613
[6]	S. Y. Lou and X. Y. Tang, “Conditional Similarity Reduction Approach: Jimbo-Miwa equation,” Chinese Physics B, Vol. 10, No. 10, 2001, p. 897.
[7]	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.
[8]	E. G. Fan, “Extended Tanh-Function Method and Its Applications to Nonlinear Equations,” Physics Letters A, Vol. 277, No. 4-5, 2000, pp. 212-218.
[9]	H. Stephen, “Differential Equations: Their Solutions Using Symmetries,” Cambridge University Press, Cambridge, 1990. http://dx.doi.org/10.1017/CBO9780511599941
[10]	M. B. Abd-el-Malek and M. M. Helal, “Characteristic Function Method for Classification of Equations of Hydrodynamics of a Perfect Luid,” Journal of Computational and Applied Mathematics, Vol. 182, No. 1, 2005, pp. 105-116.
[11]	L. J. F. Broer, “Approximate Equations for Long Water Waves,” Applied Scientific Research, Vol. 31, No. 5, 1975, pp. 377-395. http://dx.doi.org/10.1007/BF00418048
[12]	D. J. Kaup, “Finding Eigenvalue Problems for Solving Nonlinear Evolution Equations,” Progress of Theoretical Physics, Vol. 54, No. 1, 1975, pp. 72-78.
[13]	L. Martinez, “Schrodinger Spectral Problems with Energy-Dependent Potentials as Sources of Nonlinear Hamiltonian Evolution Equations,” Journal of Mathematical Physics, Vol. 21, No. 9, 1980, pp. 2342-2349. http://dx.doi.org/10.1063/1.524690
[14]	B. A. Kupershmidt, “Mathematics of Dispersive Water Waves,” Communications in Mathematical Physics, Vol. 99, No. 1, 1985, pp. 51-73. http://dx.doi.org/10.1007/BF01466593
[15]	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.
[16]	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 Naturforschung A, Vol. 61, No. 1, 2007, pp. 249-252.
[17]	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.
[18]	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.
[19]	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 Naturforschung A, Vol. 62, No. 1, 2008, pp. 121-126.
[20]	B. Q. Li, S. Li and Y. L. Ma, “New Exact Solution and Novel Time Solitons for the Dissipative Zabolotskaya Khokhlov Equation from Nonlinear Acoustics,” Zeitschrift für Naturforschung A, Vol. 67, No. 11, 2012, pp. 601-607. http://dx.doi.org/10.5560/ZNA.2011-0063