Solitary-wave solutions to a dual equation of the Kaup-Boussinesq system

In this paper, we employ the bifurcation theory of planar dynamical systems to investigate the travelling-wave solutions to a dual equation of the Kaup-Boussinesq system. The expressions for smooth solitary-wave solutions are obtained.

[1]  Mikhail V. Pavlov,et al.  Prolifiration scheme for Kaup-Boussinesq system , 2001 .

[2]  Dazhao Lü,et al.  Jacobi elliptic function solutions for two variant Boussinesq equations , 2005 .

[3]  袁玉波,et al.  Bifurcations of travelling wave solutions in variant Boussinesq equations , 2006 .

[4]  Zheng-rong Liu,et al.  A note on solitary waves for modified forms of Camassa–Holm and Degasperis–Procesi equations , 2007 .

[5]  Jianwei Shen,et al.  Bifurcations of smooth and non-smooth travelling wave solutions in the generalized Camassa–Holm equation , 2005 .

[6]  Gennady El,et al.  Integrable Shallow‐Water Equations and Undular Bores , 2001 .

[7]  Jonatan Lenells,et al.  Traveling wave solutions of the Degasperis-Procesi equation , 2005 .

[8]  Allen Parker,et al.  Cusped solitons of the Camassa–Holm equation. II. Binary cuspon–soliton interactions , 2009 .

[9]  E. J. Parkes,et al.  Some periodic and solitary travelling-wave solutions of the short pulse equation , 2008 .

[10]  A. Parker.  Wave dynamics for peaked solitons of the Camassa–Holm equation , 2008 .

[11]  Ji-Huan He,et al.  Construction of solitary solution and compacton-like solution by variational iteration method , 2006 .

[12]  Emmanuel Yomba,et al.  The extended Fan's sub-equation method and its application to KdV¿MKdV, BKK and variant Boussinesq equations , 2005 .

[13]  Vladimir B. Matveev,et al.  Solutions presque périodiques et a N-solitons de l'équation hydrodynamique non linéaire de Kaup , 1979 .

[14]  Mohammad Taghi Darvishi,et al.  New solitary wave and periodic solutions of the foam drainage equation using the Exp-function method , 2009 .

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

[16]  Allen Parker,et al.  Cusped solitons of the Camassa-Holm equation. I. Cuspon solitary wave and antipeakon limit , 2007 .

[17]  Hong-qing Zhang,et al.  New explicit and exact travelling wave solutions for a system of variant Boussinesq equations in mathematical physics , 1999 .

[18]  Tifei Qian,et al.  Peakons and periodic cusp waves in a generalized Camassa–Holm equation , 2001 .

[19]  Jibin Li,et al.  TRAVELING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR DISPERSIVE EQUATIONS , 2002 .

[20]  Lixin Tian,et al.  Soliton solution of the osmosis K(2,2)K(2,2) equation , 2008, 0907.5522.

[21]  Zhang Jiefang,et al.  Multi-solitary wave solutions for variant boussinesq equations and kupershmidt equations , 2000 .

[23]  Darryl D. Holm,et al.  An integrable shallow water equation with peaked solitons. , 1993, Physical review letters.

[24]  Adrian Constantin,et al.  Stability of the Camassa-Holm solitons , 2002, J. Nonlinear Sci..

[25]  A. O. Smirnov.  Real finite-gap regular solutions of the Kaup-Boussinesq equation , 1986 .

[26]  Luo Dingjun,et al.  Bifurcation Theory and Methods of Dynamical Systems , 1998 .

[27]  Engui Fan,et al.  Two new applications of the homogeneous balance method , 2000 .

[28]  E. J. Parkes,et al.  Periodic and solitary-wave solutions of the Degasperis-Procesi equation , 2004 .

[29]  R. Johnson,et al.  On solutions of the Camassa-Holm equation , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[30]  P. Guha.  Geodesic Flow on Extended Bott-Virasoro Group and Generalized Two Component Peakon Type Dual Systems , 2008 .

[31]  Mingliang Wang.  SOLITARY WAVE SOLUTIONS FOR VARIANT BOUSSINESQ EQUATIONS , 1995 .

[32]  Wentao Huang,et al.  Bifurcations of travelling wave solutions for the Gilson–Pickering equation , 2009 .

[33]  Lixin Tian,et al.  Solitons, peakons and periodic cusp wave solutions for the Fornberg-Whitham equation , 2010 .

[34]  Y. C. Hon,et al.  A series of travelling wave solutions for two variant Boussinesq equations in shallow water waves , 2003 .

[35]  Li-Qun Chen,et al.  Peakons and periodic cusp wave solutions in a generalized Camassa–Holm equation , 2006 .

[36]  A. M. Kamchatnov,et al.  Asymptotic soliton train solutions of Kaup–Boussinesq equations , 2003 .

[37]  D. J. Kaup,et al.  Finding Eigenvalue Problems for Solving Nonlinear Evolution Equations , 1975 .

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