Jacobi Elliptic Function Solutions for (2 + 1) Dimensional Boussinesq and Kadomtsev-Petviashvili Equation

Author(s)
Chunhuan Xiang

ABSTRACT

(2 + 1) dimensional Boussinesq and Kadomtsev-Petviashvili equation are investigated by employing Jacobi elliptic function expansion method in this paper. As a result, some new forms traveling wave solutions of the equation are reported. Numerical simulation results are shown. These new solutions may be important for the explanation of some practical physical problems. The results of this paper show that Jacobi elliptic function method can be a useful tool in obtaining evolution solutions of nonlinear system.

(2 + 1) dimensional Boussinesq and Kadomtsev-Petviashvili equation are investigated by employing Jacobi elliptic function expansion method in this paper. As a result, some new forms traveling wave solutions of the equation are reported. Numerical simulation results are shown. These new solutions may be important for the explanation of some practical physical problems. The results of this paper show that Jacobi elliptic function method can be a useful tool in obtaining evolution solutions of nonlinear system.

KEYWORDS

Jacobi Elliptic Function, Traveling Wave Solution, Kadomtsev-Petviashvili Equation, Jacobi Elliptic Function Expansion Method, Numerical Simulation

Jacobi Elliptic Function, Traveling Wave Solution, Kadomtsev-Petviashvili Equation, Jacobi Elliptic Function Expansion Method, Numerical Simulation

Cite this paper

nullC. Xiang, "Jacobi Elliptic Function Solutions for (2 + 1) Dimensional Boussinesq and Kadomtsev-Petviashvili Equation,"*Applied Mathematics*, Vol. 2 No. 11, 2011, pp. 1313-1316. doi: 10.4236/am.2011.211183.

nullC. Xiang, "Jacobi Elliptic Function Solutions for (2 + 1) Dimensional Boussinesq and Kadomtsev-Petviashvili Equation,"

References

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[2] J. L. Zhang and Y. M. Wang, “Exact Solutions to Two Nonlinear Equations,” Acta Physica Sinca, Vol. 52, 2003, pp. 1574-1578.

[3] W. L. Chan and X. Zhang, “Symmetries, Conservation Laws and Hamiltonian Structures of the Non-Isospectral and Variable Coefficient KdV and MKdV Equations,” Journal of Physics A: Mathematical and General, Vol. 28, No. 2, 1995, pp. 407-412. doi:10.1088/0305-4470/28/2/016

[4] Z. N. Zhu, “Soliton-Like Solutions of Generalized KdV Equation with External Force Term,” Acta Physica Sinca, Vol. 41, 1992, pp. 1561-1566.

[5] C. Xiang, “Analytical Solutions of KdV Equation with Relaxation Effect of Inhomogeneous Medium,” Applied Mathematics and Computation, Vol. 216, No. 8, 2010, pp. 2235-2239. doi:10.1016/j.amc.2010.03.077

[6] T. Brugarino and P. Pantano, “The Integration of Burgers and Korteweg-de Vries Equations with Nonuniformities,” Physics Letters A, Vol. 80, No. 4, 1980, pp. 223-224. doi:10.1016/0375-9601(80)90005-5

[7] C. Tian and L. G. Redekopp, “Symmetries and a Hierarchy of the General KdV Equation,” Journal of Physics A: Mathematical and General, Vol. 20, No. 2, 1987, pp. 359-366. doi:10.1088/0305-4470/20/2/021

[8] J. F. Zhang and F. Y. Chen, “Truncated Expansion Method and New Exact Soliton-Like Solution of the General Variable Coeffcient KdV Equation,” Acta Physica Sinca, Vol. 50, 2001, pp. 1648-1650.

[9] D. S. Li and H. Q. Zhang, “Improved Tanh-Function Method and the New Exact Solutions for the General Variable Coefficient KdV Equation and MKdV Equation,” Acta Physica Sinca, Vol. 52, 2003, pp. 1569-1573.

[10] Z. Y. Yan and H. Q. Zhang, “Exact Soliton Solutions of the Variable Coefficient KdV-MKdV Equation with Three Arbitrary Functions,” Acta Physica Sinca, Vol. 48, 1999, pp. 1957-1961.

[11] Z. T. Fu, S. D. Liu and S. K. Liu, “New Jacobi Ellitic Function Expansion and New Periodic Solutions of Nonlinear Wave Equations,” Physics Letters A, Vol. 290, No. 1-2, 2001, pp. 72-76. doi:10.1016/S0375-9601(01)00644-2

[12] F. J. Chen and J. F. Zhang, “Soliton-Like Solution for the (2 + 1)-Dimensional Variable Coefficient Kadomtsev-Petviashvili Equation,” Acta Armamentar, Vol. 24, 2003, pp. 389-391.

[13] M. Malfliet and E. Wieers, “A Nonlinear Theory of Charged-Particle Stopping in Non-Ideal Plasmas,” Journal of Plasma Physics, Vol. 56, No. 3, 1996, pp. 441-443. doi:10.1017/S0022377800019401

[14] E. Date, M. Jimbo and M. Kashiwara, “Transformation Groups for Soliton Equations, IV. A New Hierarchy of Soliton Equations of KP Type,” Physica D: Nonlinear Phenomena, Vol. 4, No. 3, 1982, pp. 343-365. doi:10.1016/0167-2789(82)90041-0

[15] M. Jimbo and T. Miwa, “Solitons and Infinite-Dimen-Sional Lie Algebras,” Publications of the Research Institute for Mathematical Sciences, Vol. 19, No. 3, 1983, pp. 943-1001.

[16] B. Zheng, “Travelling Wave Solutions of Two Nonlinear Evolution Equations by Using The G’/G-Expansion Method,” Applied Mathematics and Computation, 2010. doi:10.1016/j.mac.2010.12.052.

[1] Gegenhasi and X. B. Hu, “A (2 + 1)-Dimensional Sinh-Gordon Equation and Its Pfaffian Generalization,” Physics Letters A, Vol. 360, No. 3, 2007, pp. 439-447. doi:10.1016/j.physleta.2006.07.031

[2] J. L. Zhang and Y. M. Wang, “Exact Solutions to Two Nonlinear Equations,” Acta Physica Sinca, Vol. 52, 2003, pp. 1574-1578.

[3] W. L. Chan and X. Zhang, “Symmetries, Conservation Laws and Hamiltonian Structures of the Non-Isospectral and Variable Coefficient KdV and MKdV Equations,” Journal of Physics A: Mathematical and General, Vol. 28, No. 2, 1995, pp. 407-412. doi:10.1088/0305-4470/28/2/016

[4] Z. N. Zhu, “Soliton-Like Solutions of Generalized KdV Equation with External Force Term,” Acta Physica Sinca, Vol. 41, 1992, pp. 1561-1566.

[5] C. Xiang, “Analytical Solutions of KdV Equation with Relaxation Effect of Inhomogeneous Medium,” Applied Mathematics and Computation, Vol. 216, No. 8, 2010, pp. 2235-2239. doi:10.1016/j.amc.2010.03.077

[6] T. Brugarino and P. Pantano, “The Integration of Burgers and Korteweg-de Vries Equations with Nonuniformities,” Physics Letters A, Vol. 80, No. 4, 1980, pp. 223-224. doi:10.1016/0375-9601(80)90005-5

[7] C. Tian and L. G. Redekopp, “Symmetries and a Hierarchy of the General KdV Equation,” Journal of Physics A: Mathematical and General, Vol. 20, No. 2, 1987, pp. 359-366. doi:10.1088/0305-4470/20/2/021

[8] J. F. Zhang and F. Y. Chen, “Truncated Expansion Method and New Exact Soliton-Like Solution of the General Variable Coeffcient KdV Equation,” Acta Physica Sinca, Vol. 50, 2001, pp. 1648-1650.

[9] D. S. Li and H. Q. Zhang, “Improved Tanh-Function Method and the New Exact Solutions for the General Variable Coefficient KdV Equation and MKdV Equation,” Acta Physica Sinca, Vol. 52, 2003, pp. 1569-1573.

[10] Z. Y. Yan and H. Q. Zhang, “Exact Soliton Solutions of the Variable Coefficient KdV-MKdV Equation with Three Arbitrary Functions,” Acta Physica Sinca, Vol. 48, 1999, pp. 1957-1961.

[11] Z. T. Fu, S. D. Liu and S. K. Liu, “New Jacobi Ellitic Function Expansion and New Periodic Solutions of Nonlinear Wave Equations,” Physics Letters A, Vol. 290, No. 1-2, 2001, pp. 72-76. doi:10.1016/S0375-9601(01)00644-2

[12] F. J. Chen and J. F. Zhang, “Soliton-Like Solution for the (2 + 1)-Dimensional Variable Coefficient Kadomtsev-Petviashvili Equation,” Acta Armamentar, Vol. 24, 2003, pp. 389-391.

[13] M. Malfliet and E. Wieers, “A Nonlinear Theory of Charged-Particle Stopping in Non-Ideal Plasmas,” Journal of Plasma Physics, Vol. 56, No. 3, 1996, pp. 441-443. doi:10.1017/S0022377800019401

[14] E. Date, M. Jimbo and M. Kashiwara, “Transformation Groups for Soliton Equations, IV. A New Hierarchy of Soliton Equations of KP Type,” Physica D: Nonlinear Phenomena, Vol. 4, No. 3, 1982, pp. 343-365. doi:10.1016/0167-2789(82)90041-0

[15] M. Jimbo and T. Miwa, “Solitons and Infinite-Dimen-Sional Lie Algebras,” Publications of the Research Institute for Mathematical Sciences, Vol. 19, No. 3, 1983, pp. 943-1001.

[16] B. Zheng, “Travelling Wave Solutions of Two Nonlinear Evolution Equations by Using The G’/G-Expansion Method,” Applied Mathematics and Computation, 2010. doi:10.1016/j.mac.2010.12.052.