Some New Exact Traveling Wave Solutions for the Generalized Benney-Luke (GBL) Equation with Any Order

Affiliation(s)

Mathematics Department, Simao Teachers’ College, Pu’er, China.

Mathematics Department, Northwest University, Xi’an, China.

Mathematics Department, Simao Teachers’ College, Pu’er, China.

Mathematics Department, Northwest University, Xi’an, China.

ABSTRACT

In the paper, a auxiliary equation expansion method and its algorithm is proposed by studying a second order nonlinear ordinary differential equation with a four-degree term. The method is applied to the generalized Ben-ney-Luke (GBL) equation with any order. As a result, some new exact traveling wave solutions are obtained which singular solutions, triangular periodic wave solutions and jacobian elliptic function solutions .This algorithm can also be applied to other nonlinear wave equations in mathematical physics.

In the paper, a auxiliary equation expansion method and its algorithm is proposed by studying a second order nonlinear ordinary differential equation with a four-degree term. The method is applied to the generalized Ben-ney-Luke (GBL) equation with any order. As a result, some new exact traveling wave solutions are obtained which singular solutions, triangular periodic wave solutions and jacobian elliptic function solutions .This algorithm can also be applied to other nonlinear wave equations in mathematical physics.

Cite this paper

J. Wang, L. Wang and K. Yang, "Some New Exact Traveling Wave Solutions for the Generalized Benney-Luke (GBL) Equation with Any Order,"*Applied Mathematics*, Vol. 3 No. 4, 2012, pp. 309-314. doi: 10.4236/am.2012.34046.

J. Wang, L. Wang and K. Yang, "Some New Exact Traveling Wave Solutions for the Generalized Benney-Luke (GBL) Equation with Any Order,"

References

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[2] M. Wadati, “Wave Propagation in Nonlinear Lattice. I,” Journal of the Physical Society of Japan, Vol. 38, No. 3, 1975, pp. 673-680. doi:10.1143/JPSJ.38.673

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[4] V. B. Matveev and M. A. Salle, “Darboux Transformation and Solitons,” Springer, Berlin, 1991.

[5] C. H. Gu, H. S. Hu and Z. X. Zhou, “Darboux Transformation in Solition Thory and Its Geometric Applications,” Shanghai Scientific & Techincal Publishers, Shanghai, 1999.

[6] C. Rogers and W. K. Schief, “B?cklund and Darboux Transformation, Geometry and Modern Application in Soliton,” Cambridge University Press, Cambridge, 2002. doi:10.1017/CBO9780511606359

[7] R. Hirota, “The Direct Method in Soliton Theroy,” Cambridge University Press, Cambidge, 2004.

[8] P. J. Olver, “Applications of Lie Groups to Differential Equations,” Springer, New York, 1993. doi:10.1007/978-1-4612-4350-2

[9] G. W. Bluman and S. Kumei “Symmetries and Differential Equations,” Springer, Berlin, 1989.

[10] J. R. Quintero, “Solitons and Periodic Traveling Waves for the 2D-Generalized Benney-Luke Equation,” Applicable Analysis, Vol. 86, No. 3, 2007, pp. 331-351. doi:10.1080/00036810601152390

[1] M. J. Ablowitz and P. A. Clarkson, “Solution, Nonlinearn Evolution Equations and Inverse Scateing,” Cambridge University Press, Cambridge, 1991. doi:10.1017/CBO9780511623998

[2] M. Wadati, “Wave Propagation in Nonlinear Lattice. I,” Journal of the Physical Society of Japan, Vol. 38, No. 3, 1975, pp. 673-680. doi:10.1143/JPSJ.38.673

[3] M. Wadati, H. Sanuki and K. Konno, “A Generalization of Inverse Scattering Method,” Journal of the Physical Society of Japan, Vol. 46, No. 6, 1975, pp. 1965-1966. doi:10.1143/JPSJ.46.1965

[4] V. B. Matveev and M. A. Salle, “Darboux Transformation and Solitons,” Springer, Berlin, 1991.

[5] C. H. Gu, H. S. Hu and Z. X. Zhou, “Darboux Transformation in Solition Thory and Its Geometric Applications,” Shanghai Scientific & Techincal Publishers, Shanghai, 1999.

[6] C. Rogers and W. K. Schief, “B?cklund and Darboux Transformation, Geometry and Modern Application in Soliton,” Cambridge University Press, Cambridge, 2002. doi:10.1017/CBO9780511606359

[7] R. Hirota, “The Direct Method in Soliton Theroy,” Cambridge University Press, Cambidge, 2004.

[8] P. J. Olver, “Applications of Lie Groups to Differential Equations,” Springer, New York, 1993. doi:10.1007/978-1-4612-4350-2

[9] G. W. Bluman and S. Kumei “Symmetries and Differential Equations,” Springer, Berlin, 1989.

[10] J. R. Quintero, “Solitons and Periodic Traveling Waves for the 2D-Generalized Benney-Luke Equation,” Applicable Analysis, Vol. 86, No. 3, 2007, pp. 331-351. doi:10.1080/00036810601152390