Bifurcation method and traveling wave solution to Whitham-Broer-Kaup equation

By using bifurcation method to Whitham-Broer-Kaup shallow water equations, bifurcation parameter sets are shown. Numbers of solitary waves, kink waves and periodic waves are given. Under various parameter conditions, all explicit formulas of solitary wave solutions, kink wave solutions and periodic wave solutions are listed.

[1]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[2]  G. B. Whitham,et al.  Variational methods and applications to water waves , 1967, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[3]  Yong Xu,et al.  Travelling wave solutions in the generalized Hirota-Satsuma coupled KdV system , 2005, Appl. Math. Comput..

[4]  Jibin Li,et al.  Smooth and non-smooth traveling waves in a nonlinearly dispersive equation , 2000 .

[5]  D. J. Kaup,et al.  A Higher-Order Water-Wave Equation and the Method for Solving It , 1975 .

[6]  P. J. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[7]  Andrew G. Glen,et al.  APPL , 2001 .

[8]  J. Rogers.  Chaos , 1876 .

[9]  Zhenya Yan,et al.  Explicit and exact traveling wave solutions of Whitham–Broer–Kaup shallow water equations , 2001 .

[10]  Paul F. Byrd,et al.  Handbook of elliptic integrals for engineers and scientists , 1971 .

[11]  Ljf Lambert Broer.  Approximate equations for long water waves , 1975 .

[12]  Jianwei Shen,et al.  Travelling wave solutions in a model of the helix polypeptide chains , 2004 .

[13]  Qi Wang,et al.  A generalized method and general form solutions to the Whitham–Broer–Kaup equation , 2004 .

[14]  B. Kupershmidt,et al.  Mathematics of dispersive water waves , 1985 .

[15]  B. N. Rolov,et al.  Method of bifurcation of nonlinear integral equations in the theory of ferroelectricity , 1976 .

[16]  L. Perko.  Differential Equations and Dynamical Systems , 1991 .