N-Fold Darboux Transformation for a Nonlinear Evolution Equation


In this paper, we present a N-fold Darboux transformation (DT) for a nonlinear evolution equation. Comparing with other types of DTs, we give the relationship between new solutions and the trivial solution. The DT presented in this paper is more direct and universal to obtain explicit solutions.

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Y. Zhao, "N-Fold Darboux Transformation for a Nonlinear Evolution Equation," Applied Mathematics, Vol. 3 No. 8, 2012, pp. 943-948. doi: 10.4236/am.2012.38141.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] M. J. Ablowitz and H. Segur, “Solitons and Inverse Scattering Transform (SIAM Studies in Applied Mathematics, No. 4),” Society for Industrial Mathematics, 2000.
[2] C. Rogers and W. K. Schief, “B?cklund and Darboux Transformations Geometry and Modern Application in Soliton Theory,” Cambridge University Press, Cambridge, 2002. doi:10.1017/CBO9780511606359
[3] C. H. Gu, H. S. Hu and Z. X. Zhou, “Darboux Transformations in Integrable Systems: Theory and Their Applications to Geometry (Mathematical Physics Studies 26),” Springer-Verlag, New York, 2005.
[4] A. R. Chowdhury, “Painlevé Analysis and Its Applications (Chapman and Hall/ CRC Monographs and Surveys in Pure and Applied Mathematics 105),” Chapman and Hall/CRC, Boca Raton, 1999.
[5] A. M. Wazwaz, “Exact Solutions of Compact and Noncompact Structures for the KP-BBM Equation,” Applied Mathematics and Computation, Vol. 169, No. 1, 2005, pp. 700-712. doi:10.1016/j.amc.2004.09.061
[6] J. H. Li, M. Jia and S. Y. Lou, “Kac-Moody-Virasoro Symmetry Algebra and Symmetry Reductions of the Bilinear Sinh-Gordon Equation in (2+1)-dimensions,” Journal of Physics A: Mathematical and Theoretical, Vol. 40, No. 7, 2007, pp. 1585-1595. doi:10.1088/1751-8113/40/7/010
[7] S. L. Zhang, S. Y. Lou and C. Z. Qu, “Functional Variable Separation for Extended (1+2)-dimensional Nonlinear Wave Equations,” Chinese Physics Letters, Vol. 22, No. 11, 2005, pp. 2731-2734. doi:10.1088/0256-307X/22/11/001
[8] Gegenhasi, X. B. Hu and H. Y. Wang, “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
[9] X. J. Liu and Y. B. Zeng, “On the Ablowitz-Ladik Equations with Self-consistent Sources,” Journal of Physics A: Mathematical and Theoretical, Vol. 40, No. 30, 2007, pp. 8765-8790. doi:10.1088/1751-8113/40/30/011
[10] S. Y. Lou, M. Jia, X. Y. Tang and F. Huang, “Vortices, Circumfluence, Symmetry Groups, and Darboux Transformations of the (2+1)-dimensional Euler Equation,” Physical Review E, Vol. 75, No. 5, 2007, Article ID: 056318. doi:10.1103/PhysRevE.75.056318
[11] C. L. Chen, Y. S. Li and J. E. Zhang, “The Multi-Soliton Solutions of the CH-γ Equation,” Science in China Series A, Vol. 51, No. 2, 2008, pp. 314-320. doi:10.1007/s11425-007-0137-x
[12] L. J. Zhou, “Darboux Transformation for the Nonisospectral AKNS System,” Physics Letters A, Vol. 345, No. 4-6, 2005, pp. 314-322. doi:10.1016/j.physleta.2005.07.046
[13] H. C. Hu, “Analytical Positon, Negaton and Complexiton Solutions for the Coupled Modified KdV System,” Journal of Physics A: Mathematical and Theoretical, Vol. 42, No. 18, 2009, Article ID: 185207. doi:10.1088/1751-8113/42/18/185207
[14] A. H. Chen and X. M. Li, “Soliton Solutions of the Coupled Dispersionless Equation,” Physics Letters A, Vol. 370, No. 3-4, 2007, pp. 281-286. doi:10.1016/j.physleta.2007.05.107
[15] J. Wang, “Darboux Transformation and Soliton Solutions for the Boiti-Pempinelli-Tu (BPT) Hierarchy,” Journal of Physics A: Mathematical and General, Vol. 38, No. 39, 2005, pp. 8367-8377. doi:10.1088/0305-4470/38/39/005
[16] L. Luo, “Darboux Transformation and Exact Solutions for A Hierarchy of Nonlinear Evolution Equations,” Journal of Physics A: Mathematical and Theoretical, Vol. 40, No. 15, 2007, pp. 4169-4179. doi:10.1088/1751-8113/40/15/008
[17] Y.Wang, L. J. Shen and D. L. Du, “Darboux Transformation and Explicit Solutions for Some (2+1)-dimensional Equations,” Physics Letters A, Vol. 366, No. 3, 2007, pp. 230-240. doi:10.1016/j.physleta.2007.02.043
[18] X. G. Geng and H. W. Tam, “Darboux Transformation and Soliton Solutions for Generalized Nonlinear Schr-?dinger Equations,” Journal of the Physical Society of Japan, Vol. 68, No. 5, 1999, pp. 1508-1512. doi:10.1143/JPSJ.68.1508
[19] X. M. Li and A. H. Chen, “Darboux Transformation and Multi-soliton Solutions of Boussinesq-Burgers Equation,” Physics Letters A, Vol. 342, No. 5-6, 2005, pp. 413-420. doi:10.1016/j.physleta.2005.05.083
[20] H. X. Wu, Y. B. Zeng and T. Y. Fan, “Complexitons of the Modified KdV Equation by Darboux Transformation,” Applied Mathematics and Computation, Vol. 196, No. 2, 2008, pp. 501-510. doi:10.1016/j.amc.2007.06.011
[21] J. S. He, L. Zhang, Y. Cheng and Y. S. Li, “Determinant Representation of Darboux Transformation for the AKNS System,” Science in China Series A, Vol. 49, No. 12, 2006, pp. 1867-1878. doi:10.1007/s11425-006-2025-1
[22] Z. Y. Yan and H. Q. Zhang, “A Lax Integrable Hierarchy, N-Hamiltonian Structure, r-Matrix, Finite-Dimensional Liouville Integrable Involutive Systems, and Involutive Solutions,” Chaos Solitons and Fractals, Vol. 13, No. 7, 2002, pp. 1439-1450. doi:10.1016/S0960-0779(01)00150-3
[23] E. Mjolhus and J. Wyller, “Nonlinear Alfvén Waves in a Finite-Beta Plasma,” Journal of Plasma Physics, Vol. 40, No. 2, 1988, pp. 299-318. doi:10.1017/S0022377800013295

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