Stability of peakons for the Degasperis‐Procesi equation

The Degasperis-Procesi equation can be derived as a member of a oneparameter family of asymptotic shallow water approximations to the Euler equations with the same asymptotic accuracy as that of the CamassaHolm equation. In this paper, we study the orbital stability problem of the peaked solitons to the Degasperis-Procesi equation on the line. By constructing a Liapunov function, we prove that the shapes of these peakon solitons are stable under small perturbations.

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

[2]  A. Bressan,et al.  Global Conservative Solutions of the Camassa–Holm Equation , 2007 .

[3]  Joachim Escher,et al.  Shock waves and blow-up phenomena for the periodic Degasperis-Procesi equation , 2007 .

[4]  Giuseppe Maria Coclite,et al.  Numerical schemes for computing discontinuous solutions of the Degasperis–Procesi equation , 2007 .

[5]  R. Johnson,et al.  Camassa–Holm, Korteweg–de Vries and related models for water waves , 2002, Journal of Fluid Mechanics.

[6]  Hans Lundmark,et al.  Multi-peakon solutions of the Degasperis–Procesi equation , 2003, nlin/0503033.

[7]  Zhaoyang Yin,et al.  On the Cauchy problem for an integrable equation with peakon solutions , 2003 .

[8]  Yoshimasa Matsuno,et al.  Multisoliton solutions of the Degasperis–Procesi equation and their peakon limit , 2005 .

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

[10]  Terence Tao,et al.  Low-regularity global solutions to nonlinear dispersive equations , 2002 .

[11]  Darryl D. Holm,et al.  Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary pde , 2002, nlin/0203007.

[12]  A. Constantin,et al.  Orbital stability of solitary waves for a shallow water equation , 2001 .

[13]  Gerard Misio łek.  A shallow water equation as a geodesic flow on the Bott-Virasoro group , 1998 .

[14]  Darryl D. Holm,et al.  S I ] 1 2 M ay 2 00 2 A New Integrable Equation with Peakon Solutions , 2002 .

[15]  Giuseppe Maria Coclite,et al.  On the well-posedness of the Degasperis-Procesi equation , 2006 .

[16]  W. Strauss,et al.  Stability of peakons , 2000 .

[17]  Athanassios S. Fokas,et al.  Symplectic structures, their B?acklund transformation and hereditary symmetries , 1981 .

[18]  L. Chambers.  Linear and Nonlinear Waves , 2000, The Mathematical Gazette.

[19]  Tosio Kato,et al.  Quasi-linear equations of evolution, with applications to partial differential equations , 1975 .

[20]  Darryl D. Holm,et al.  Wave Structure and Nonlinear Balances in a Family of Evolutionary PDEs , 2002, SIAM J. Appl. Dyn. Syst..

[21]  Zhaoyang Yin,et al.  Global solutions to a new integrable equation with peakons , 2004 .

[22]  Joachim Escher,et al.  Global weak solutions and blow-up structure for the Degasperis–Procesi equation , 2006 .

[23]  Thomas Kappeler,et al.  On geodesic exponential maps of the Virasoro group , 2007 .

[24]  A. Constantin.  Existence of permanent and breaking waves for a shallow water equation: a geometric approach , 2000 .

[25]  Hans Lundmark,et al.  Formation and Dynamics of Shock Waves in the Degasperis-Procesi Equation , 2007, J. Nonlinear Sci..

[26]  J. Escher,et al.  Wave breaking for nonlinear nonlocal shallow water equations , 1998 .

[27]  J. Escher,et al.  Global existence and blow-up for a shallow water equation , 1998 .

[28]  Darryl D. Holm,et al.  An integrable shallow water equation with linear and nonlinear dispersion. , 2001, Physical review letters.

[29]  Zhaoyang Yin,et al.  Global Existence and Blow-Up Phenomena for the Degasperis-Procesi Equation , 2006 .

[30]  A. Awane,et al.  k‐symplectic structures , 1992 .

[31]  D. Henry.  Infinite propagation speed for the Degasperis–Procesi equation☆ , 2005 .

[32]  Antonio Degasperis,et al.  A test in Asymptotic Integrability of 1 + 1 wave equations , 1998 .

[33]  A. Constantin,et al.  Geodesic flow on the diffeomorphism group of the circle , 2003 .

[34]  Darryl D. Holm,et al.  A New Integrable Equation with Peakon Solutions , 2002, nlin/0205023.

[35]  O. Mustafa.  A Note on the Degasperis-Procesi Equation , 2005 .