Solution of KdV and Boussinesq using Darboux Transformation

Year 2018, Volume: 3 Issue: 3, 16 - 27, 31.12.2018

Mohamed R. Ali

Abstract

Two Darboux transformations of the Korteweg-de Vries (KdV) equation and Boussinesq equation are constructed through
the Darboux method. Soliton solutions of these two equations are presented by applying the Darboux transformations.

Keywords

Darboux transformation, multisoliton solution, spectral problem, generalized Korteweg-de Vries equations

References

• [1] V. B. Matveev and M. A. Salle, Darboux Transformation and Solitons. Springer. (1991).
• [2] C. Rogers and W. K. Schief, B¨acklund and Darboux transformations: geometry and modern applications in soliton theory, Cambridge Texts in Applied Mathematics. (2002).
• [3] A. A. Halim, Korteweg-de-Vries equations in problems of fluid dynamic. (2001) 1-10.
• [4] P.G. Estevez and J .Prada, Singular manifold method for an equation in 2 + 1 dimensions, Journal of Nonlinear Mathematical Physics. 12 (2005) 266–279.
• [5] J.Weiss, M. Tabor and G. Carnevale, The Painlev´e property for partial differential equations, J. Math. Phys. 24 (1983) 522-526.
• [6] P.G. Estevez and P.R. Gordoa, The Singular Manifold Method: Darboux Transformations and Nonclassical Symmetries, Nonlinear Mathematical Physics. 2 (1995) 334–355.
• [7] G. Chaohao, H. Hesheng and Z. Zixiang, Darboux Transformations In Integrable Systems Theory And Their Applications To Geometry, Institute of Mathematics, Fudan University, Shanghai, China. (2005) PP. 2.
• [8] L.Wazzan, A modifed tanh–coth method for solving the KdV and the KdV–Burgers’ equations, Communications in Nonlinear Science and Numerical Simulation .14 (2009) 443–450.
• [9] G. Chaohao, H. Hesheng and Z. Zixiang, Darboux Transformations In Integrable Systems Theory And Their Applications To Geometry, Institute of Mathematics, Fudan University, Shanghai, China.(2005) PP. 85.
• [10] R. Sadat, M. Kassem, Explicit Solutions for the (2+ 1)-Dimensional Jaulent–Miodek Equation Using the Integrating Factors Method in an Unbounded Domain. Mathematical and Computational Applications, 23(1)(2018) 1-9.
• [11] M. A. Ramadan, M. R. Ali, An effcient hybrid method for solving fredholm integral equations using triangular functions, New Trends in Mathematical Sciences,5(1) (2017) 213-224.
• [12] M. A. Ramadan, M. R. Ali, Numerical Solution of Volterra-Fredholm Integral Equations Using Hybrid Orthonormal Bernstein and Block-Pulse Functions, Asian Research Journal of Mathematics,4(4) (2017) 1-14.
• [13] M. A. Ramadan, M. R. Ali, Application of Bernoulli wavelet method for numerical solution of fuzzy linear Volterra-Fredholm integral equations, Communication in Mathematical Modeling and Applications,2(3)(2017) 40-49.
• [14] M. A. Ramadan, M. R. Ali, Solution of integral and Integro-Differential equations system using Hybrid orthonormal Bernstein and block-pulse functions, Journal of abstract and computational mathematics,2(1)(2017) 35-48.