Darboux Transformation and New Multi-Soliton Solutions of the Whitham-Broer-Kaup System


Through a variable transformation, the Whitham-Broer-Kaup system is transformed into a parameter Levi system. Based on the Lax pair of the parameter Levi system, the N-fold Darboux transformation with multi-parameters is constructed. Then some new explicit solutions for the Whitham-Broer-Kaup system are obtained via the given Darboux transformation.

Share and Cite:

Xu, T. (2015) Darboux Transformation and New Multi-Soliton Solutions of the Whitham-Broer-Kaup System. Applied Mathematics, 6, 20-27. doi: 10.4236/am.2015.61003.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Zabusky, N.J. and Galvin, C.J. (1971) Shallow-Water Waves, the Korteweg-de Vries Equation and Solitons. Journal of Fluid Mechanics, 47, 811-824.
[2] Dullin, H.R., Georg, A.G. and Holm, D.D. (2003) Camassa-Holm, Korteweg-de Vries-5 and Other Asymptotically Equivalent Equations for Shallow Water Waves. Fluid Dynamics Research, 33, 73-95.
[3] Chakravarty, S. and Kodama, Y. (2009) Soliton Solutions of the KP Equation and Application to Shallow Water Waves. Studies in Applied Mathematics, 123, 83-151.
[4] Kodama, Y. (2010) KP Solitons in Shallow Water. Journal of Physical A: Mathematical and Theoretical, 43, Article ID: 434004.
[5] Lambert, F., Musette, M. and Kesteloot, E. (1987) Soliton Resonances for the Good Boussinesq Equation. Inverse Problems, 3, 275-288.
[6] Li, Y.S. and Zhang, J.E. (2001) Darboux Transformation of Classical Boussinesq System and Its Multi-Soliton Solutions. Physics Letters A, 284, 253-258.
[7] Ablowitz, M.J. and Clarkson, P.A. (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge.
[8] Rogers, C. and Schief, W.K. (2002) B?cklund and Darboux Transformations Geometry and Modern Application in Soliton Theory. Cambridge University Press, Cambridge.
[9] Hirota, R. (2004) The Direct Method in Soliton Theory. Cambridge University Press, Cambridge.
[10] Gu, C.H., Hu, H.S. and Zhou, Z.X. (2005) Darboux Transformation in Soliton Theory and Its Geometric Applications. Shanghai Science Technology Publication House, Shanghai.
[11] Whitham, G.B. (1967) Variational Methods and Applications to Water Wave. Proceedings of the Royal Society A, 299, 6-25.
[12] Broer, L.J. (1975) Approximate Equations for Long Water Waves. Applied Scientific Research, 31, 377-395.
[13] Kaup, D.J. (1975) A Higher-Order Water Equation and Method for Solving It. Progress of Theoretical Physics, 54, 396-408.
[14] Kupershmidt, B.A. (1985) Mathematics of Dispersive Water Waves. Communications in Mathematical Physics, 99, 51-73.
[15] Xia, Z. (2004) Homogenous Balance Method and Exact Analytical Solutions for Whitham-Broer-Kaup Equations in the Shallow Water. Chinese Quarterly Journal of Mathematics, 19, 240-246.
[16] Xie, F.D. and Gao, X.S. (2004) A Computational Approach to the New Type Solutions of Whitham-Broer-Kaup Equation in Shallow Water. Communications in Theoretical Physics, 41, 179-182.
[17] Zhang, J.F., Guo, G.P. and Wu, F.M. (2002) New Multi-Soliton Solutions and Travelling Wave Solutions of the Dispersive Long-Wave Equations. Chinese Physics, 11, 533-536.
[18] Lin, G.D., Gao, Y.T., Gai, X.L. and Meng, D.X. (2011) Extended Double Wronskian Solutions to the Whitham-Broer-Kaup Equations in Shallow Water. Nonlinear Dynamics, 64, 197-206.
[19] Wang, L., Gao, Y.T. and Gai, X.T. (2012) Gauge Transformation, Elastic and Inelastic Interactions for the Whitham-Broer-Kaup Shallow-Water Model. Communications in Nonlinear Science and Numerical Simulation, 17, 2833-2844.
[20] Geng, X.G. and Tam, H.W. (1999) Darboux Transformation and Soliton Solutions for Generalized Nonlinear Schrödinger Equations. Journal of Physical Society of Japan, 68, 1508-1512.
[21] Huang, D.J. and Zhang, H.Q. (2008) Vandermonde-Like Determinants’ Representations of Darboux Transformations and Expliclt Solutions for the Modified Kadomtsev-Petviashvili Equation. Physica A, 387, 4565-4580.

Copyright © 2023 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.