Solution of KdV and Boussinesq using Darboux Transformation

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. [1] V. B. Matveev and M. A. Salle, Darboux Transformation and Solitons. Springer. (1991).
2. [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. [3] A. A. Halim, Korteweg-de-Vries equations in problems of fluid dynamic. (2001) 1-10.
4. [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. [5] J.Weiss, M. Tabor and G. Carnevale, The Painlev´e property for partial differential equations, J. Math. Phys. 24 (1983) 522-526.
6. [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. [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. [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. [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. [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. [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. [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. [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. [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.