Non-Traveling Wave Solutions for the (1 + 1)-Dimensional Burgers System by Riccati Equation Mapping Approach

ABSTRACT

Starting from the symbolic computation system Maple and Riccati equation mapping approach and a linear variable separation approach, a new family of non-traveling wave solutions of the (1 + 1)-dimensional Burgers system is derived.

Cite this paper

R. Xu and S. Ma, "Non-Traveling Wave Solutions for the (1 + 1)-Dimensional Burgers System by Riccati Equation Mapping Approach,"*Applied Mathematics*, Vol. 4 No. 10, 2013, pp. 123-125. doi: 10.4236/am.2013.410A3015.

R. Xu and S. Ma, "Non-Traveling Wave Solutions for the (1 + 1)-Dimensional Burgers System by Riccati Equation Mapping Approach,"

References

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http://dx.doi.org/10.1088/0256-307X/19/6/308

[2] 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. 6, 1997, Article ID: 6401. http://dx.doi.org/10.1063/1.532219

[3] 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.

http://dx.doi.org/10.1103/PhysRevE.66.046601

[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, Article ID: 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.

http://dx.doi.org/10.1088/1009-1963/10/10/303

[7] 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.

http://dx.doi.org/10.1016/S0960-0779(02)00636-7

[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.

http://dx.doi.org/10.1016/S0375-9601(00)00725-8

[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.

http://dx.doi.org/10.1016/j.cam.2004.11.042

[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/10.1016/j.cam.2004.11.042

[12] D. J. Kaup, “Finding Eigenvalue Problems for Solving Nonlinear Evolution Equations,” Progress of Theoretical Physics, Vol. 54, No. 1, 1975, pp. 72-78.

http://dx.doi.org/10.1143/PTP.54.72

[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] 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.

http://dx.doi.org/10.4236/am.2012.31002

[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. 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.

[18] 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.

[19] 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.

[1] S. Y. Lou and X. Y. Tang, “Fractal Solutions of the Nizhnik-Novikov-Veselov Equation,” Chinese Physics Letters, Vol. 19, No. 6, 2002, pp. 769-771.

http://dx.doi.org/10.1088/0256-307X/19/6/308

[2] 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. 6, 1997, Article ID: 6401. http://dx.doi.org/10.1063/1.532219

[3] 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.

http://dx.doi.org/10.1103/PhysRevE.66.046601

[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, Article ID: 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.

http://dx.doi.org/10.1088/1009-1963/10/10/303

[7] 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.

http://dx.doi.org/10.1016/S0960-0779(02)00636-7

[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.

http://dx.doi.org/10.1016/S0375-9601(00)00725-8

[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.

http://dx.doi.org/10.1016/j.cam.2004.11.042

[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/10.1016/j.cam.2004.11.042

[12] D. J. Kaup, “Finding Eigenvalue Problems for Solving Nonlinear Evolution Equations,” Progress of Theoretical Physics, Vol. 54, No. 1, 1975, pp. 72-78.

http://dx.doi.org/10.1143/PTP.54.72

[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] 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.

http://dx.doi.org/10.4236/am.2012.31002

[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. 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.

[18] 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.

[19] 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.