The Characteristic Function Method and Its Application to (1 + 1)-Dimensional Dispersive Long Wave Equation

ABSTRACT

In this paper, the characteristic function method is applied to seek traveling wave solutions of nonlinear partial differential equations in a unified way. We consider the Wu-Zhang equation (which describes (1 + 1)-dimensional disper-sive long wave). The equations governing the wave propagation consist of a pair of non linear partial differential equations. The characteristic function method reduces the system of nonlinear partial differential equations to a system of nonlinear ordinary differential equations which is solved via the shooting method, coupled with Rungekutta scheme. The results include kink-profile solitary wave solutions, periodic wave solutions and rational solutions. As an illustrative example, the properties of some soliton solutions for Wu-Zhang equation are shown by some figures.

In this paper, the characteristic function method is applied to seek traveling wave solutions of nonlinear partial differential equations in a unified way. We consider the Wu-Zhang equation (which describes (1 + 1)-dimensional disper-sive long wave). The equations governing the wave propagation consist of a pair of non linear partial differential equations. The characteristic function method reduces the system of nonlinear partial differential equations to a system of nonlinear ordinary differential equations which is solved via the shooting method, coupled with Rungekutta scheme. The results include kink-profile solitary wave solutions, periodic wave solutions and rational solutions. As an illustrative example, the properties of some soliton solutions for Wu-Zhang equation are shown by some figures.

Cite this paper

M. Helal, M. Mekky and E. 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. doi: 10.4236/am.2012.31002.

M. Helal, M. Mekky and E. Mohamed, "The Characteristic Function Method and Its Application to (1 + 1)-Dimensional Dispersive Long Wave Equation,"

References

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[2] V. A. Matveev and M. A. Salle, “Darboux Transformations and Solitons,” Springer-Verlag, Berlin, Heidelberg, 1991.

[3] E. G. Fan, “Extended Tanh-Function Method and Its Applications to Nonlinear Equations,” Physics Letters A, Vol. 277, No. 4-5, 2000, pp. 212-2188. doi:10.1016/S0375-9601(00)00725-8

[4] Y. B. Zhou, M. L. Wang and Y. M. Wang, “Periodic Wave Solutions to a Coupled KdV Equations with Variable Coefficients,” Physics Letters A, Vol. 308, No. 1, 2003, pp. 31-36.

[5] P. J. Olever, “Applications of Lie Groups to Differential Equations,” Springer, New York, 1968.

[6] G. W. Bluman and S. Kumei, “Symmetries and Differential Equations,” Springer, New York, 1989.

[7] H. Stephen,” Differential Equations: Their Solutions Using Symmetries,” Cambridge University Press, Cambridge, 1990. doi:10.1017/CBO9780511599941

[8] N. H. Ibragimov, “CRC Handbook of Lie Group Analysis of Differential Equation,” CRC Press, Boca Raton, 1996.

[9] M. L. Wang, Y. B. Zhou and Z. B. Li, “Application of a Homogeneous Balance Method to Exact Solutions of Nonlinear Equations in Mathematical Physics,” Physics Letters A, Vol. 216, No. 1-5, 1996, pp. 67-75. doi:10.1016/0375-9601(96)00283-6

[10] E. G. Fan and J. Zhang,” Applications of Jacobi Elliptic Function Method to Special-Type Nonlinear Equations,” Physics Letters A, Vol. 305, No. 6, 2002, pp.383-392. doi:10.1016/S0375-9601(02)01516-5

[11] R. Seshadri and T. Y. Na, “Group Invariance in Engineering Boundary Value Problems,” Springer-Verlag, New York, 1985. doi:10.1007/978-1-4612-5102-6

[12] 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. doi:10.1016/j.cam.2004.11.042

[13] M. B. Abd-el-Malek and M. M. Helal, “The Characteristic Function Method and Exact Solutions of Nonlinear Sheared Flows with Free Surface under Gravity,” Journal of Computational and Applied Mathematics, Vol. 189, No. 1-2, 2006, pp. 2-21. doi:10.1016/j.cam.2005.04.038

[14] T. Y. Wu and J. E. Zhang, “On Modeling Nonlinear Long Wave,” In: L. P. Cook, V. Roytbhurd and M. Tulin, Eds., Mathematics Is for Solving Problems, Society for Industrial and Applied Mathematics, Philadelphia, 1996, p. 233.

[15] L. J. F. Broer, “Approximate Equations for Long Water Waves,” Applied Scientific Research, Vol. 31, No. 5, 1975, pp. 377-395. doi:10.1007/BF00418048

[16] D. J. Kaup, “Finding Eigenvalue Problems for Solving Nonlinear Evolution Equations,” Progress of Theoretical Physics, Vol. 54, No. 1, 1975, pp. 72-78. doi:10.1143/PTP.54.72

[17] 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. doi:10.1063/1.524690

[18] B. A. Kupershmidt, “Mathematics of Dispersive Water Waves,” Communications in Mathematical Physics, Vol. 99, No. 1, 1985, pp. 51-73. doi:10.1007/BF01466593

[19] C. L. Chen and S. Y. Lou, “Soliton Excitations and Periodic Waves without Dispersion Relation in Shallow Water System,” Chaos, Solitons & Fractals, Vol. 16, No. 1, 2003, pp. 27-35. doi:10.1016/S0960-0779(02)00148-0

[20] M. L. Wang, “Solitary Wave Solutions for Variant Boussinesq Equations,” Physics Letters A, Vol. 199, No. 3-4, 1995, pp. 169-172. doi:10.1016/0375-9601(95)00092-H

[21] X. D. Zheng, Y. Chen and H. Q. Zhang, “Generalized Extended Tanh-Function Method and Its Application to (1 + 1)-Dimensional Dispersive Long Wave Equation,” Physics Letters A, Vol. 311, No. 2-3, 2003, pp. 145-157. doi:10.1016/S0375-9601(03)00451-1

[22] X. Zeng, D. Wang and S. Wang, “A Generalized Extended Rational Expansion Method and Its Application to (1 + 1)-Dimensional Dispersive Long Wave Equation,” Applied Mathematics and Computation, Vol. 212, No. 2, 2009, pp. 296-304. doi:10.1016/j.amc.2009.02.020

[1] M. Wadati, H. Sanuki and K. Konno, “Relationships among Inverse Method, B?cklund Transformation and an Infinite Number of Conservation Laws,” Progress of Theoretical Physics, Vol. 53, No. 2, 1975, pp. 419-436. doi:10.1143/PTP.53.419

[2] V. A. Matveev and M. A. Salle, “Darboux Transformations and Solitons,” Springer-Verlag, Berlin, Heidelberg, 1991.

[3] E. G. Fan, “Extended Tanh-Function Method and Its Applications to Nonlinear Equations,” Physics Letters A, Vol. 277, No. 4-5, 2000, pp. 212-2188. doi:10.1016/S0375-9601(00)00725-8

[4] Y. B. Zhou, M. L. Wang and Y. M. Wang, “Periodic Wave Solutions to a Coupled KdV Equations with Variable Coefficients,” Physics Letters A, Vol. 308, No. 1, 2003, pp. 31-36.

[5] P. J. Olever, “Applications of Lie Groups to Differential Equations,” Springer, New York, 1968.

[6] G. W. Bluman and S. Kumei, “Symmetries and Differential Equations,” Springer, New York, 1989.

[7] H. Stephen,” Differential Equations: Their Solutions Using Symmetries,” Cambridge University Press, Cambridge, 1990. doi:10.1017/CBO9780511599941

[8] N. H. Ibragimov, “CRC Handbook of Lie Group Analysis of Differential Equation,” CRC Press, Boca Raton, 1996.

[9] M. L. Wang, Y. B. Zhou and Z. B. Li, “Application of a Homogeneous Balance Method to Exact Solutions of Nonlinear Equations in Mathematical Physics,” Physics Letters A, Vol. 216, No. 1-5, 1996, pp. 67-75. doi:10.1016/0375-9601(96)00283-6

[10] E. G. Fan and J. Zhang,” Applications of Jacobi Elliptic Function Method to Special-Type Nonlinear Equations,” Physics Letters A, Vol. 305, No. 6, 2002, pp.383-392. doi:10.1016/S0375-9601(02)01516-5

[11] R. Seshadri and T. Y. Na, “Group Invariance in Engineering Boundary Value Problems,” Springer-Verlag, New York, 1985. doi:10.1007/978-1-4612-5102-6

[12] 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. doi:10.1016/j.cam.2004.11.042

[13] M. B. Abd-el-Malek and M. M. Helal, “The Characteristic Function Method and Exact Solutions of Nonlinear Sheared Flows with Free Surface under Gravity,” Journal of Computational and Applied Mathematics, Vol. 189, No. 1-2, 2006, pp. 2-21. doi:10.1016/j.cam.2005.04.038

[14] T. Y. Wu and J. E. Zhang, “On Modeling Nonlinear Long Wave,” In: L. P. Cook, V. Roytbhurd and M. Tulin, Eds., Mathematics Is for Solving Problems, Society for Industrial and Applied Mathematics, Philadelphia, 1996, p. 233.

[15] L. J. F. Broer, “Approximate Equations for Long Water Waves,” Applied Scientific Research, Vol. 31, No. 5, 1975, pp. 377-395. doi:10.1007/BF00418048

[16] D. J. Kaup, “Finding Eigenvalue Problems for Solving Nonlinear Evolution Equations,” Progress of Theoretical Physics, Vol. 54, No. 1, 1975, pp. 72-78. doi:10.1143/PTP.54.72

[17] 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. doi:10.1063/1.524690

[18] B. A. Kupershmidt, “Mathematics of Dispersive Water Waves,” Communications in Mathematical Physics, Vol. 99, No. 1, 1985, pp. 51-73. doi:10.1007/BF01466593

[19] C. L. Chen and S. Y. Lou, “Soliton Excitations and Periodic Waves without Dispersion Relation in Shallow Water System,” Chaos, Solitons & Fractals, Vol. 16, No. 1, 2003, pp. 27-35. doi:10.1016/S0960-0779(02)00148-0

[20] M. L. Wang, “Solitary Wave Solutions for Variant Boussinesq Equations,” Physics Letters A, Vol. 199, No. 3-4, 1995, pp. 169-172. doi:10.1016/0375-9601(95)00092-H

[21] X. D. Zheng, Y. Chen and H. Q. Zhang, “Generalized Extended Tanh-Function Method and Its Application to (1 + 1)-Dimensional Dispersive Long Wave Equation,” Physics Letters A, Vol. 311, No. 2-3, 2003, pp. 145-157. doi:10.1016/S0375-9601(03)00451-1

[22] X. Zeng, D. Wang and S. Wang, “A Generalized Extended Rational Expansion Method and Its Application to (1 + 1)-Dimensional Dispersive Long Wave Equation,” Applied Mathematics and Computation, Vol. 212, No. 2, 2009, pp. 296-304. doi:10.1016/j.amc.2009.02.020