New Periodic Solitary Wave Solutions for a Variable-Coefficient Gardner Equation from Fluid Dynamics and Plasma Physics

ABSTRACT

The Gardner equation with a variable-coefficient from fluid dynamics and plasma physics is investigated. Different kinds of solutions including breather-type soliton and two soliton solutions are obtained using bilinear method and extended homoclinic test approach. The proposed method can also be applied to solve other types of higher dimensional integrable and non-integrable systems.

The Gardner equation with a variable-coefficient from fluid dynamics and plasma physics is investigated. Different kinds of solutions including breather-type soliton and two soliton solutions are obtained using bilinear method and extended homoclinic test approach. The proposed method can also be applied to solve other types of higher dimensional integrable and non-integrable systems.

KEYWORDS

Extended Homoclinic Test Approach, Bilinear Form, Gardner Equation with a Variable-Coefficient, Periodic Solitary Wave Solutions

Extended Homoclinic Test Approach, Bilinear Form, Gardner Equation with a Variable-Coefficient, Periodic Solitary Wave Solutions

Cite this paper

nullM. Abdou, "New Periodic Solitary Wave Solutions for a Variable-Coefficient Gardner Equation from Fluid Dynamics and Plasma Physics,"*Applied Mathematics*, Vol. 1 No. 4, 2010, pp. 307-311. doi: 10.4236/am.2010.14040.

nullM. Abdou, "New Periodic Solitary Wave Solutions for a Variable-Coefficient Gardner Equation from Fluid Dynamics and Plasma Physics,"

References

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[2] S. A. El-Wakil and M. A. Abdou, “New Applications of Adomian Decomposition Method,” Chaos, Solitons and Fractals, Vol. 33, No. 2, 2007, pp. 513-522.

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[13] X.-G. Xu, X. Meng, Y. Gao and X. Wen, “Analytic N- Solitary-Wave Solution of a Variable-Coefficient Gardner Equation from Fluid Dynamics and Plasma Physics,” Applied Mathematics and Computation, Vol. 210, No. 2, 2009, pp. 313-320.

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[18] E. J. Fan, “Auto-B?cklund Transformation and Similarity Reductions for General Variable Coefficient Kdv Equations,” Physics Letters A, Vol. 294, No. 1, 2002, pp. 26- 30.

[19] G. Xu, X. H. Meng, Y. T. Gao and X. Wen, “Analytic N-Solitary Wave Solution for a Variable-Coefficient Gardner Equation from Fluid Dynamics and Plasma Physics,” Applied Mathematics and Computation, Vol. 210, No. 2, 2009, pp. 313-320.

[20] Z. D. Dai, Z. J. Liu and D. L. Li, “Exact Periodic Solitary-Wave Solutions for the Kdv Equation,” Chinese Physics Letters, Vol. 25, No. 5, 2008, pp. 1531-1533.

[21] Z. D. Dai, M. R. Jiang, Q. Y. Dai and S. L. Li, “Homoclinic Bifurcation for the Boussinesq Equation with Even Constraints,” Chinese Physics Letters, Vol. 23, No. 5, 2006, pp. 1065-1067.

[22] Z. D. Dai, J. Liu and D. L. Li, “Applications of HTA and EHTA to the YTSF Equation,” Applied Mathematics and Computation, Vol. 207, No. 2, 2009, pp. 360-364.

[23] Z. D. Dai, J. Liu, X. P. Zeng and Z. J. Liu, “Periodic Kink-Wave and Kinky Periodic-Wave Solutions for the Jimbo-Miwa Equation,” Physics Letters A, Vol. 372, No. 38, 2008, pp. 5984-5986.

[24] Z. Dai, L. Song, H. Fu and X. Zeng, “Exact Three Wave Solutions for the KP Equation,” Applied Mathematics and Computation, Vol. 216, No. 5, 2010, pp. 1599-1604.

[25] C. Wang, Z. D. Dai and L. Liang, “Exact Three Wave Solution for Higher Dimensional Kdv Equation,” Applied Mathematics and Computation, Vol. 216, No. 2, 2010, pp. 501-505.

[26] W. Chuan, Z. D. Dai, M. Gui and L. S. Qing, “New Exact Periodic Solitary Wave Solutions for New (2 + 1)-Dimen- sional Kdv Equation,” Communications in Theoretical Physics, Vol. 52, No. 2, 2009, pp. 862-864.

[27] D. L. Li and J. X. Zhao, “New Exact Solutions to the (2 + 1)-Diomensional Ito Equation, Extended Homoclinic Test Technique,” Applied Mathematics and Computation, Vol. 215, No. 5, 2009, pp. 1968-1974.

[28] Z.-D. Dai, C.-J. wang, S.-Q. Lin, D. L. Li and G. Mu, “The Three Wave Method for Nonlinear Evolution Equations,” Nonlinear Science Letter A, Vol. 1, No. 5, 2010, pp. 77-82.

[1] M. Ablowitz and P. A. Clarkson, “Soliton, Nonlinear Evolution Equations and Inverse Scattering,” Cambridge University Press, New York, 1991.

[2] S. A. El-Wakil and M. A. Abdou, “New Applications of Adomian Decomposition Method,” Chaos, Solitons and Fractals, Vol. 33, No. 2, 2007, pp. 513-522.

[3] S. A. El-Wakil and M. A. Abdou, “New Exact Travelling Wave Solutions of Two Nonlinear Physical Models,” Nonlinear Analysis, Vol. 68, No. 2, 2008, pp. 235-245.

[4] J.-H. He and M. A. Abdou, “New Periodic Solutions for Nonlinear Evolution Equations Using Exp Function Method,” Chaos, Solitons and Fractals, Vol. 34, No. 5, 2007, pp. 1421-1429.

[5] M. A. Abdou and S. Zhang, “New Periodic Wave Solutions via Extended Mapping Method,” Communication in Nonlinear Science and Numerical Simulation, Vol. 14, No. 1, 2009, pp. 2-11.

[6] M. A. Abdou, “On the Variational Iteration Method,” Physics Letters A, Vol. 366, No. 1-2, 2007, pp. 61-68.

[7] R. Hirota, “Direct Method in Soliton Theory,” In: R. K. Bullough and P. J. Caudrey, Ed., Solitons, Springer, Berlin, 1980, pp. 1-196.

[8] M. A. Abdou, “Generalized Solitary and Periodic Solutions for Nonlinear Partial Differential Equations by the Exp-Function Method,” Journal of Nonlinear Dynamics, Vol. 52, No. 1-2, 2008, pp. 1-9.

[9] G. M. Wei, Y. T. Gao and X. G. Xu, “Painlevé Analysis and Transformations for a Generalized Two-Dimensional Variable-Coefficient Burgers Model from Fluid Mechanics, Acoustics and Cosmic-Ray Astrophysics,” Nuovo Cimento B, Vol. 121, No. 4, 2006, pp. 327-342.

[10] P. E. P. Holloway, E. Pelinovsky, T. Talipova and B. Barnes, “A Nonlinear Model of Internal Tide Transformation on the Australian North West Shelf,” Journal of Physical Oceanography, Vol. 27, No. 6, 1997, pp. 871- 896.

[11] J. A. Gear and R. Grimshaw, “A Second-Order Theory for Solitary Waves in Shallow Fluids,” Physics of Fluid, Vol. 26, No. 14, 1983, 16 pages.

[12] S. Watanabe, “Ion Acoustic Soliton in Plasma with Negative Ion,” Journal of Physical Society of Japan, Vol. 53, 1984, pp. 950-956.

[13] X.-G. Xu, X. Meng, Y. Gao and X. Wen, “Analytic N- Solitary-Wave Solution of a Variable-Coefficient Gardner Equation from Fluid Dynamics and Plasma Physics,” Applied Mathematics and Computation, Vol. 210, No. 2, 2009, pp. 313-320.

[14] N. Joshi, “Painlevé Property of General Variable- Coefficient Versions of the Korteweg-De Vries and Non-Linear Schr?dinger Equations,” Physics Letter A, Vol. 125, No. 9, 1987, p. 456.

[15] R. Grimshaw, Proceedings of the Royal Society of London, Series A, 1979, p. 359.

[16] Z. X. Chen, B. Y. Guo and L. W. Xiang, “Complete Integrability and Analytic Solutions of a Kdv‐Type Equation,” Journal of Mathematical Physics, Vol. 31, 1990, p. 2851.

[17] W. P. Hong and Y. D. Jung, “Auto-B?cklund Transformation and Analytic Solutions for General Variable- Coefficient Kdv Equation,” Physics Letters A, Vol. 257, No. 3-4, 1999, pp. 149-157.

[18] E. J. Fan, “Auto-B?cklund Transformation and Similarity Reductions for General Variable Coefficient Kdv Equations,” Physics Letters A, Vol. 294, No. 1, 2002, pp. 26- 30.

[19] G. Xu, X. H. Meng, Y. T. Gao and X. Wen, “Analytic N-Solitary Wave Solution for a Variable-Coefficient Gardner Equation from Fluid Dynamics and Plasma Physics,” Applied Mathematics and Computation, Vol. 210, No. 2, 2009, pp. 313-320.

[20] Z. D. Dai, Z. J. Liu and D. L. Li, “Exact Periodic Solitary-Wave Solutions for the Kdv Equation,” Chinese Physics Letters, Vol. 25, No. 5, 2008, pp. 1531-1533.

[21] Z. D. Dai, M. R. Jiang, Q. Y. Dai and S. L. Li, “Homoclinic Bifurcation for the Boussinesq Equation with Even Constraints,” Chinese Physics Letters, Vol. 23, No. 5, 2006, pp. 1065-1067.

[22] Z. D. Dai, J. Liu and D. L. Li, “Applications of HTA and EHTA to the YTSF Equation,” Applied Mathematics and Computation, Vol. 207, No. 2, 2009, pp. 360-364.

[23] Z. D. Dai, J. Liu, X. P. Zeng and Z. J. Liu, “Periodic Kink-Wave and Kinky Periodic-Wave Solutions for the Jimbo-Miwa Equation,” Physics Letters A, Vol. 372, No. 38, 2008, pp. 5984-5986.

[24] Z. Dai, L. Song, H. Fu and X. Zeng, “Exact Three Wave Solutions for the KP Equation,” Applied Mathematics and Computation, Vol. 216, No. 5, 2010, pp. 1599-1604.

[25] C. Wang, Z. D. Dai and L. Liang, “Exact Three Wave Solution for Higher Dimensional Kdv Equation,” Applied Mathematics and Computation, Vol. 216, No. 2, 2010, pp. 501-505.

[26] W. Chuan, Z. D. Dai, M. Gui and L. S. Qing, “New Exact Periodic Solitary Wave Solutions for New (2 + 1)-Dimen- sional Kdv Equation,” Communications in Theoretical Physics, Vol. 52, No. 2, 2009, pp. 862-864.

[27] D. L. Li and J. X. Zhao, “New Exact Solutions to the (2 + 1)-Diomensional Ito Equation, Extended Homoclinic Test Technique,” Applied Mathematics and Computation, Vol. 215, No. 5, 2009, pp. 1968-1974.

[28] Z.-D. Dai, C.-J. wang, S.-Q. Lin, D. L. Li and G. Mu, “The Three Wave Method for Nonlinear Evolution Equations,” Nonlinear Science Letter A, Vol. 1, No. 5, 2010, pp. 77-82.