A Super Camassa–Holm Equation with N‐Peakon Solutions

A super Camassa–Holm equation with peakon solutions is proposed, which is associated with a 3 × 3 matrix spectral problem with two potentials. With the aid of the zero-curvature equation, we derive a hierarchy of super Harry Dym type equations and establish their Hamiltonian structures. It is shown that the super Camassa–Holm equation is exactly a negative flow in the hierarchy and admits exact solutions with N peakons. As an example, exact 1-peakon solutions of the super Camassa–Holm equation are given. Infinitely many conserved quantities of the super Camassa–Holm equation and the super Harry Dym type equation are, respectively, obtained.

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

[2]  Xianguo Geng,et al.  A three-component generalization of Camassa–Holm equation with N-peakon solutions , 2011 .

[3]  Xavier Raynaud,et al.  A convergent numerical scheme for the Camassa--Holm equation based on multipeakons , 2005 .

[4]  B. Kupershmidt,et al.  A super Korteweg-de Vries equation: An integrable system , 1984 .

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

[6]  Athanassios S. Fokas,et al.  On the structure of symplectic operators and hereditary symmetries , 1980 .

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

[8]  Ziemowit Popowicz,et al.  The fully supersymmetric AKNS equations , 1990 .

[9]  Y. Manin,et al.  A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy , 1985 .

[10]  Allen Parker,et al.  The Peakon Limits of Soliton Solutions of the Camassa-Holm Equation(General) , 2006 .

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

[12]  Yoshimasa Matsuno,et al.  Parametric Representation for the Multisoliton Solution of the Camassa–Holm Equation , 2005, nlin/0504055.

[13]  A S Carstea,et al.  Extension of the bilinear formalism to supersymmetric KdV-type equations , 1998, solv-int/9812022.

[14]  Z. Popowicz.  A 2-component or N=2 supersymmetric Camassa–Holm equation , 2005, nlin/0509050.

[15]  G. Roelofs,et al.  Supersymmetric extensions of the nonlinear Schrödinger equation: Symmetries and coverings , 1992 .

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

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

[18]  Q. P. Liu.  Darboux transformations for supersymmetric korteweg-de vries equations , 1995 .

[19]  Fred Cooper,et al.  Solitons in the Camassa-Holm shallow water equation , 1993, patt-sol/9311006.

[20]  C. Yung,et al.  Hirota Bilinear Form for the Super-KdV Hierarchy , 1993 .

[21]  A S Carstea,et al.  Bilinear Approach to Supersymmetric KdV Equation , 2001 .

[22]  Hans Lundmark,et al.  Degasperis-Procesi peakons and the discrete cubic string , 2005, nlin/0503036.

[23]  Darryl D. Holm,et al.  A Class of Equations with Peakon and Pulson Solutions (with an Appendix by Harry Braden and John Byatt-Smith) , 2004, nlin/0412029.

[24]  Xianguo Geng,et al.  An extension of integrable peakon equations with cubic nonlinearity , 2009 .

[25]  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.

[26]  M. Chaichian,et al.  On the method of inverse scattering problem and Bäcklund transformations for supersymmetric equations , 1978 .

[27]  The supersymmetric Camassa–Holm equation and geodesic flow on the superconformal group , 1998, solv-int/9811016.

[28]  Jonatan Lenells,et al.  On the N =2 supersymmetric Camassa-Holm and Hunter-Saxton equations , 2008, 0809.0077.

[29]  Helge Holden,et al.  Real-valued algebro-geometric solutions of the Camassa–Holm hierarchy , 2001, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

[31]  Liu,et al.  Asymptotic Integrability of Water Waves. , 1996, Physical review letters.

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

[33]  Jingsong He,et al.  A supertrace identity and its applications to superintegrable systems , 2008 .

[34]  On the bi-Hamiltonian structures of the Camassa-Holm and Harry Dym equations , 2004, nlin/0407057.

[35]  Yishen Li,et al.  The multiple-soliton solution of the Camassa-Holm equation , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[36]  Jing Ping Wang,et al.  Integrable peakon equations with cubic nonlinearity , 2008, 0805.4310.

[37]  K. Ikeda.  A supersymmetric extension of the Toda lattice hierarchy , 1987 .

[38]  Allan P. Fordy,et al.  Coupled Harry Dym equations with multi-Hamiltonian structures , 1988 .

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

[40]  V. Gerdjikov,et al.  Inverse scattering transform for the Camassa–Holm equation , 2006, Inverse Problems.

[41]  Darryl D. Holm,et al.  A New Integrable Shallow Water Equation , 1994 .

[42]  Metin Gürses,et al.  A super AKNS scheme , 1985 .

[43]  M. Gürses,et al.  A super soliton connection , 1986 .

[44]  Helge Holden,et al.  Real-valued algebro-geometric solutions of the Camassa-Holm hierarchy. , 2008, Philosophical transactions. Series A, Mathematical, physical, and engineering sciences.

[45]  J. F. Gomes,et al.  Deformations of N = 2 superconformal algebra and supersymmetric two-component Camassa–Holm equation , 2006, hep-th/0611192.

[46]  Q. P. Liu,et al.  Darboux transformation for the Manin-Radul supersymmetric KdV equation , 1997 .

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

[48]  Xing-Biao Hu,et al.  An approach to generate superextensions of integrable systems , 1997 .

[49]  S. Stanciu.  Additional symmetries of supersymmetric KP hierarchies , 1993, hep-th/9309058.

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

[51]  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.

[52]  Pierre Mathieu,et al.  Supersymmetric extension of the Korteweg--de Vries equation , 1988 .