Periodic Waves and their Limits for the Camassa-holm equation

In this paper, the bifurcation method of dynamical systems is employed to study the Camassa–Holm equation \[ u_t+2ku_{x}-u_{xxt}+auu_x=2u_{x}u_{xx}+uu_{xxx}. \] We investigate the periodic wave solutions of form u = φ(ξ) which satisfy φ(ξ + T) = φ(ξ), here ξ = x - ct and c, T are constants. Their six implicit expressions and two explicit expressions are obtained. We point out that when the initial values are changed, the periodic waves may become three waves, periodic cusp waves, smooth solitary waves and peakons. Our results give an explanation to the appearance of periodic cusp waves and peakons. Moreover, three sets of graphs of the implicit functions are drawn, and three sets of numerical simulations are displayed. The identity of these graphs and simulations imply the correctness of our theoretical results.

[1]  Zheng-rong Liu,et al.  Peakons of the Camassa–Holm equation , 2002 .

[2]  Pascal Redou,et al.  On the Inverse Scattering Approach to the Camassa-Holm Equation , 2003, math-ph/0403039.

[3]  Darryl D. Holm,et al.  On asymptotically equivalent shallow water wave equations , 2003, nlin/0307011.

[4]  J. Boyd.  Peakons and coshoidal waves: traveling wave solutions of the Camassa-Holm equation , 1997 .

[5]  Zhengrong Liu,et al.  New Bounded Traveling Waves of Camassa-holm equation , 2004, Int. J. Bifurc. Chaos.

[6]  A. Constantin.  On the scattering problem for the Camassa-Holm equation , 2001, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[7]  Z. Yin.  On the Blow-Up Scenario for the Generalized Camassa–Holm Equation , 2004 .

[8]  H. Dai,et al.  Solitary shock waves and other travelling waves in a general compressible hyperelastic rod , 2000, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[9]  J. Lenells.  Conservation laws of the Camassa–Holm equation , 2005 .

[10]  Min-ying Tang,et al.  Extension on peaked wave solutions of CH-γ equation , 2004 .

[11]  P. Olver,et al.  Well-posedness and Blow-up Solutions for an Integrable Nonlinearly Dispersive Model Wave Equation , 2000 .

[12]  Discrete Lagrangian systems on the Virasoro group and Camassa–Holm family , 2002, math-ph/0209037.

[13]  Henrik Kalisch,et al.  Stability of solitary waves for a nonlinearly dispersive equation , 2004 .

[14]  Z. Yin.  On the Cauchy Problem for a Nonlinearly Dispersive Wave Equation , 2003, math/0309183.

[15]  A. Constantin.  Finite propagation speed for the Camassa-Holm equation , 2005 .

[16]  Tifei Qian,et al.  Peakons and their bifurcation in a generalized Camassa-Holm equation , 2001, Int. J. Bifurc. Chaos.

[17]  Zheng-rong Liu,et al.  Peaked wave solutions of Camassa–Holm equation , 2004 .

[18]  W. Strauss,et al.  Stability of a class of solitary waves in compressible elastic rods , 2000 .

[19]  H. McKean.  Breakdown of the Camassa‐Holm equation , 2004 .

[20]  M. Klein,et al.  The Marchenko–Ostrovski mapping and the trace formula for the Camassa–Holm equation , 2003 .

[21]  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.

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

[23]  H. Dai.  Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod , 1998 .

[24]  H. P. McKean,et al.  Fredholm determinants and the Camassa‐Holm hierarchy , 2003 .

[25]  J. Lenells.  The Scattering Approach for the Camassa–Holm equation , 2002, nlin/0306021.

[26]  Wenling Zhang.  General expressions of peaked traveling wave solutions of CH-γ and CH equations , 2004 .

[27]  H. McKean.  The Liouville correspondence between the Korteweg—de Vries and the Camassa‐Holm hierarchies , 2003 .

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

[29]  R. Danchin.  A note on well-posedness for Camassa-Holm equation , 2003 .

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