Persistence properties and unique continuation for a generalized Camassa-Holm equation

In this paper, persistence properties of solutions are investigated for a generalized Camassa-Holm equation (g-k bCH) having (k+1)-degree nonlinearities and containing as its integrable members the Camassa-Holm, the Degasperis-Procesi, and the Novikov equations. The persistence properties will imply that strong solutions of the g-k bCH equation will decay at infinity in the spatial variable provided that the initial data does. Furthermore, it is shown that the equation exhibits unique continuation for appropriate values of the parameters b and k. Finally, existence of global solutions is established when b = k+1.

[1]  Feride Tiglay,et al.  The periodic Cauchy problem for Novikov's equation , 2010, 1009.1820.

[2]  J. Lenells.  Traveling wave solutions of the Camassa-Holm equation , 2005 .

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

[4]  Jerry L. Bona,et al.  Solutions of the Korteweg-de Vries equation in frac-tional order Sobolev space , 1976 .

[5]  D. Henry.  Compactly Supported Solutions of the Camassa-Holm Equation , 2005 .

[6]  A. Alexandrou Himonas,et al.  The Cauchy problem for the Novikov equation , 2012 .

[7]  Darryl D. Holm,et al.  Traveling Wave Solutions for a Class of One-Dimensional Nonlinear Shallow Water Wave Models , 2004 .

[8]  Yong Zhou,et al.  Persistence Properties and Unique Continuation of Solutions of the Camassa-Holm Equation , 2006 .

[9]  A. Himonas,et al.  On well-posedness of the Degasperis-Procesi equation , 2011 .

[10]  Z. Qiao.  Communications in Mathematical Physics The Camassa-Holm Hierarchy , N-Dimensional Integrable Systems , and Algebro-Geometric Solution on a Symplectic Submanifold , 2003 .

[11]  Adrian Constantin,et al.  A shallow water equation on the circle , 1999 .

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

[13]  C. Kenig,et al.  On Uniqueness Properties of Solutions of Schrödinger Equations , 2006 .

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

[15]  Herbert Koch,et al.  Nonlinear wave interactions for the Benjamin-Ono equation. , 2004 .

[16]  Terence Tao,et al.  Sharp global well-posedness for KdV and modified KdV on ℝ and , 2003 .

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

[18]  S. Hakkaev,et al.  On the Cauchy problem for the periodic b-family of equations and of the non-uniform continuity of Degasperis-Procesi equation , 2009 .

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

[20]  R. S. Johnson,et al.  The Camassa–Holm equation for water waves moving over a shear flow , 2003 .

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

[22]  J. Bourgain,et al.  Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations , 1993 .

[23]  Hans Lundmark,et al.  Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm type equation , 2009, 0903.3663.

[24]  A. Constantin,et al.  The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations , 2007, 0709.0905.

[25]  A. Himonas,et al.  Norm Inflation and Ill-Posedness for the Degasperis-Procesi Equation , 2014 .

[26]  Richard Beals,et al.  Multipeakons and the Classical Moment Problem , 1999, solv-int/9906001.

[27]  On uniqueness properties of solutions of the k-generalized KdV equations , 2006, math/0601621.

[28]  G. Ponce,et al.  Non-uniform continuity in $H\sp 1$ of the solution map of the CH equation , 2007 .

[29]  W. Strauss,et al.  Gain of regularity for equations of KdV type , 1992 .

[30]  Jerry L. Bona,et al.  Sharp well-posedness results for the BBM equation , 2008 .

[31]  Luis Vega,et al.  A bilinear estimate with applications to the KdV equation , 1996 .

[32]  Vladimir S. Novikov,et al.  Generalizations of the Camassa–Holm equation , 2009 .

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

[34]  Vladimir S. Novikov,et al.  Perturbative symmetry approach , 2002, nlin/0203055.

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

[36]  A. Alexandrou Himonas,et al.  Non-Uniform Dependence on Initial Data of Solutions to the Euler Equations of Hydrodynamics , 2010 .

[37]  H. McKean.  Breakdown of a shallow water equation , 1998 .

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