跳至主要内容

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

Read  full  paper  at:
http://www.scirp.org/journal/PaperInformation.aspx?PaperID=52956#.VK3oWMnQrzE

Author(s) 

ABSTRACT
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.
 
Cite this paper
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.
 
References
[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.
http://dx.doi.org/10.1017/S0022112071001393
 
[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.
http://dx.doi.org/10.1016/S0169-5983(03)00046-7
 
[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.
http://dx.doi.org/10.1111/j.1467-9590.2009.00448.x
 
[4]Kodama, Y. (2010) KP Solitons in Shallow Water. Journal of Physical A: Mathematical and Theoretical, 43, Article ID: 434004.
http://dx.doi.org/10.1088/1751-8113/43/43/434004
 
[5]Lambert, F., Musette, M. and Kesteloot, E. (1987) Soliton Resonances for the Good Boussinesq Equation. Inverse Problems, 3, 275-288.
http://dx.doi.org/10.1088/0266-5611/3/2/010
 
[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.
http://dx.doi.org/10.1016/S0375-9601(01)00331-0
 
[7]Ablowitz, M.J. and Clarkson, P.A. (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511623998
 
[8]Rogers, C. and Schief, W.K. (2002) B?cklund and Darboux Transformations Geometry and Modern Application in Soliton Theory. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511606359
 
[9]Hirota, R. (2004) The Direct Method in Soliton Theory. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9780511543043
 
[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.
http://dx.doi.org/10.1098/rspa.1967.0119
 
[12]Broer, L.J. (1975) Approximate Equations for Long Water Waves. Applied Scientific Research, 31, 377-395.
http://dx.doi.org/10.1007/BF00418048
 
[13]Kaup, D.J. (1975) A Higher-Order Water Equation and Method for Solving It. Progress of Theoretical Physics, 54, 396-408.
http://dx.doi.org/10.1143/PTP.54.396
 
[14]Kupershmidt, B.A. (1985) Mathematics of Dispersive Water Waves. Communications in Mathematical Physics, 99, 51-73.
http://dx.doi.org/10.1007/BF01466593
 
[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.
http://dx.doi.org/10.1088/0253-6102/41/2/179
 
[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.
http://dx.doi.org/10.1088/1009-1963/11/6/303
 
[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.
http://dx.doi.org/10.1007/s11071-010-9857-5
 
[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.
http://dx.doi.org/10.1016/j.cnsns.2011.11.018
 
[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.
http://dx.doi.org/10.1143/JPSJ.68.1508
 
[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.                                                                                                                     eww150108lx
标签：

评论

发表评论

此博客中的热门博文