Share This Article:

Pulse Soliton Solutions of the Modified KdV and Born-Infeld Equations

Full-Text HTML XML Download Download as PDF (Size:143KB) PP. 135-140
DOI: 10.4236/ijmnta.2013.22017    4,888 Downloads   7,487 Views   Citations


In this work, we use the Bogning-Djeumen Tchaho-Kofané method to look for all solutions of shape Sechn- of the modified KdV and Born-Infeld Equations. n being a real number, we obtain the soliton solutions when n is positive and the non soliton solutions when n is negative.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

J. Bogning, "Pulse Soliton Solutions of the Modified KdV and Born-Infeld Equations," International Journal of Modern Nonlinear Theory and Application, Vol. 2 No. 2, 2013, pp. 135-140. doi: 10.4236/ijmnta.2013.22017.


[1] R. Hirota, “The Direct Method in Soliton Theory,” Cambridge University Press, Cambridge, 2004. doi:10.1017/CBO9780511543043
[2] W. Herman and A. Nuseir, “Symbolic Methods to Construct Exact Solutions of Nonlinear Partial Differential Equations,” Mathematics and Computers in Simulation, Vol. 43, No. 1, 1997, pp. 13-27. doi:10.1016/S0378-4754(96)00053-5
[3] A. M. Wazwaz, “Soliton Solutions for Two (3 + 1)-Dimensional Non Integrable KdV-Type Equations,” Mathematical and Computer Modelling, Vol. 55, No. 5-6, 2012, pp. 1845-1848. doi:10.1016/j.mcm.2011.11.082
[4] A. M. Wazwaz, “Two Forms of (3 + 1)-Dimensional B-Type Kadomtsev-Petviashvili Equation: Multiple Soliton Solutions,” Physica Scripta, Vol. 86, No. 3, 2012, pp. 035007-035015. doi:10.1088/0031-8949/86/03/035007
[5] Q. C. Jiang, Y. L. Su and X. M. Ji, “Coupling Effects of Grey-Grey Separate Spatial Screening Soliton Pairs,” Physica Scripta, Vol. 86, No. 3, 2012, pp. 035404-035408. doi:10.1088/0031-8949/86/03/035404
[6] T. Kaladze, S. Mahmood and H. Ur-Rehman, “Acoustic Nonlinear Periodic (Cnoidal) Waves and Solitons in Pair-Ion Plasmas,” Physica Scripta, Vol. 86, No. 3, 2012, pp. 035506-035514. doi:10.1088/0031-8949/86/03/035506
[7] Q.-Y. Chen, P. G. Kevrekidis and B. A. Malomed, “Dynamics of Bright Solitons and Soliton Arrays in the Nonlinear Schrodinger Equation with a Combination of Random and Harmonic Potentials,” Physica Scripta, Vol. 2012, No. T149, 2012, pp. 014001-014007. doi:10.1088/0031-8949/2012/T149/014001
[8] M. Born and L. Infeld, “1934 Foundations of the New Field Theory,” Proceedings of the Royal Society A, Vol. 144, No. 852, 1934, pp. 425-451.
[9] M. Born, “On the Quantum Theory of Electromagnetic Field,” Proceedings of the Royal Society A, Vol. 143, No. 849, 1934, pp. 410-437.
[10] D. I. Blokhintsev, “Space and Time in the Microcosm,” Nanka, Moscow, 1982.
[11] M. KoiV and V. Rosenhaus, “Family of Two Dimensional Born-Infeld Equations and a System of Conservation Laws,” IZV. Akad. Nauk Est. SSR. Fizika, MathematiKa, Vol. 28, No. 3, 1979, pp. 187-193.
[12] B. M. Barbashov and N. A. Chernikov, “Solving and Quantization of Nonlinear Two-Dimensional Model Born-Infeld Type,” Zhurn EKsperin I Teor Fiziki, Vol. 60 No. 5, 1966, pp. 1926-1308.
[13] B. M. Barbashov and N. A. Chernikov, “Interaction of Two Plane Waves in Born-Infeld Electrodynamics,” Fizika Vysokikh Energii I Teoria Elementarnykh Chastitz, Kyiv, 1967, pp. 733-743.
[14] W. I. Fushchych and V. A. Tychinin, “On Linearization of Some Nonlinear Equations with the Help of Non Local Transformations,” Institute of Mathematics, Academic of Sciences, Ukraine, 1982.
[15] V. Fedorchuk, “Symmetry Reduction and Exact Solutions of the Euler-Lagrange-Born-Infeld, Multidimensional Monge-Ampere and Eikonal Equations,” Journal of Nonlinear Mathematical Physics, Vol. 2, No. 3-4, 1995, pp. 329-333.
[16] R. K. Bullough, “The Wave Par Excellence: The Solitary Progressive Great Wave of Equilibrium of the Fluid and Early History of the Solitary Wave,” Springer, New York, 1988, pp. 150-281.
[17] J. de Frutos and J. M. Sanz-Serna, “Accuracy and Conservation Properties in Numerical Integration. The Case of the Korteweg-de Vries Equation,” Numerische Mathematik, Vol. 75, No. 4, 1997, pp. 421-445. doi:10.1007/s002110050247
[18] J. S. Russel, “Report on Waves: Report of the 14th Meeting of the British Association for the Advancement of Science,” York, 1884, pp. 314-390.
[19] W. E. Schiesser, “Method of Lines Solution of the Korteweg-de Vries Equation,” Computers & Mathematics with Applications, Vol. 28, No. 10-21, 1994, pp. 147-154.
[20] E. Varley and B. R. Seymour “A Simple Derivation of the N-Soliton Solutions to the Korteweg—de Vries Equation,” SIAM: SIAM Journal on Applied Mathematics, Vol. 58, No. 3, 1998, pp. 904-911. doi:10.1137/S0036139996303270
[21] D. Vvedenskii, “Partial Differential Equations with Mathematica,” Wokingham Addison, Wesley, 1992.
[22] S. Wolfram, “The Mathematica Book. Cambridge (UK),” Cambridge University Press, 1999.
[23] A. M. Wazwaz, “A Modified KdV-Type Equation That Admits a Variety of Travelling Wave Solutions: Kinks, Solitons, Peakons and Cuspons,” Physica Scripta, Vol. 86, No. 4, 2012, pp. 045501-045506. doi:10.1088/0031-8949/86/04/045501
[24] C. T. Djeumen Tchaho, J. R. Bogning and T. C. Kofane, “TC 2010 Construction of the Analytical Solitary Wave Solutions of Modified Kuramoto-Sivashinsky Equation by the Method of Identification of Coefficients of the Hyperbolic Functions,” Far East Journal of Dynamical Systems, Vol. 14, No. 1, 2010, pp. 14-17.
[25] C. T. Djeumen Tchaho, J. R. Bogning and T. C. Kofane, “Multi-Soliton Solutions of the Modified Kuramoto-Sivashinsky’s Equation by the BDK Method,” Far East Journal of Dynamical Systems, Vol. 15, No. 2, 2011, pp. 83-98.
[26] J. R. Bogning, C. T. Djeumen Tchaho and T. C. Kofané, “Construction of the Soliton Solutions of the Ginzburg-Landau Equations by the New Bogning-Djeumen Tchaho-Kofané method,” Physica Scripta, Vol. 85, No. 2, 2012, pp. 025013-025017. doi:10.1088/0031-8949/85/02/025013
[27] C. T. Djeumen Tchaho , J. R. Bogning, and T. C. Kofané, “Modulated Soliton Solution of the Modified Kuramoto-Sivashinsky’s Equation,” American Journal of Computational and Applied Mathematics, Vol. 2, No. 5, 2012, pp. 218-224. doi:10.5923/j.ajcam.20120205.03
[28] J. R. Bogning, C. T. Djeumen Tchaho and T. C. Kofané, “TC 2012 Generalization of the Bogning-Djeumen Tchaho-Kofane Method for the Construction of the Solitary Waves and the Survey of the Instabilities,” Far East Journal of Dynamical Systems, Vol. 20, No. 2, 2012, pp. 101-119.

comments powered by Disqus

Copyright © 2018 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.