Advertisement

Letters in Mathematical Physics

, Volume 75, Issue 1, pp 1–15 | Cite as

A Two-component Generalization of the Camassa-Holm Equation and its Solutions

  • Ming Chen
  • Si-Qi liu
  • Youjin Zhang
Article

Abstract

An explicit reciprocal transformation between a two-component generalization of the Camassa–Holm equation, called the 2-CH system, and the first negative flow of the AKNS hierarchy is established. This transformation enables one to obtain solutions of the 2-CH system from those of the first negative flow of the AKNS hierarchy. Interesting examples of peakon and multi-kink solutions of the 2-CH system are presented

Keywords

Camassa–Holm equation AKNS hierarchy reciprocal transformation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abenta S., Grava T. Modulation of Camassa–Holm equation and reciprocal transformations. math-ph/0506042Google Scholar
  2. 2.
    Ablowitz M.J., Kaup D.J., Newell A.C., Segur H. (1974). The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math. 53:249–315MathSciNetGoogle Scholar
  3. 3.
    Alber M.S., Camassa R., Fedorov Yu.N., Holm D.D., Marsden J.E. (2001). The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDEs of shallow water and Dym type. Comm. Math. Phys. 221:197–227zbMATHCrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Antonowicz M., Fordy A.P. (1987). Coupled K dV equations with multi-Hamiltonian structures. Physica D28:345–357zbMATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Antonowicz M., Fordy A.P. (1988). Coupled Harry Dym equations with multi-Hamiltonian structures. J. Phys. A21:L269–L275CrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Antonowicz M., Fordy A.P. (1989). Factorisation of energy dependent Schrödinger operators: Miura maps and modified systems. Comm. Math. Phys. 124:465–486zbMATHCrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Beals R., Sattinger D.H., Szmigielski J. (2000). Multipeakons and the classical moment problem. Adv. Math. 154:229–257zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Camassa R., Holm D.D. (1993). An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71:1661–1664zbMATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Camassa R., Holm D.D., Hyman J.M. (1994). A new integrable shallow water equation. Adv. Appl. Mech. 31:1–33Google Scholar
  10. 10.
    Constantin A. (2001). On the scattering problem for the Camassa–Holm equation. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457:953–970zbMATHADSMathSciNetGoogle Scholar
  11. 11.
    Constantin A., McKean H.P. (1999). A shallow water equation on the circle. Comm. Pure Appl. Math. 52:949–982CrossRefMathSciNetGoogle Scholar
  12. 12.
    Constantin A., Strauss W.A. (2002). Stability of the Camassa–Holm solitons. J. Nonlinear Sci. 12:415–422zbMATHCrossRefADSMathSciNetGoogle Scholar
  13. 13.
    Constantin A., Strauss W.A. (2000). Stability of peakons. Comm. Pure Appl. Math. 53:603–610zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Degasperis, A., Holm, D.D., Hone, A.N.W.: Integrable and non-integrable equations with peakons. In: Nonlinear physics: theory and experiment, II (Gallipoli, 2002), pp. 37–43, World Scientific, River Edge, (2003).Google Scholar
  15. 15.
    Dubrovin, B., Zhang, Y.: Normal forms of integrable PDEs, Frobenius manifolds and Gromov–Witten invariants. math.DG/0108160Google Scholar
  16. 16.
    Dubrovin, B., Liu, S.Q., Zhang, Y.: On Hamiltonian perturbations of hyperbolic systems of conservation laws, I: quasi-triviality of bi-Hamiltonian perturbations. Commun. Pure Appl. Math. (to appear) math.DG/0410027Google Scholar
  17. 17.
    Falqui, G., On a two-component generalization of the CH equation. In: Talk given at the conference “Analytic and geometric theory of the Camassa–Holm equation and Integrable systems”, Bologna (2004)Google Scholar
  18. 18.
    Fokas A.S. (1995). On a class of physically important integrable equations. Physica D87:145–150ADSMathSciNetGoogle Scholar
  19. 19.
    Fuchssteiner B. (1996). Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa–Holm equation. Physica D95:229–243zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Fuchssteiner B., Fokas A.S. (1981). Symplectic structures, their Bäcklund transformations and hereditary symmetries. Physica D4:47–66CrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Gu C.H., Zhou Z.X. (1987). On the Darboux matrices of Bäcklund transformations for AKNS systems. Lett. Math. Phys. 13:179–187zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Hone A.N.W. (1999). The associated Camassa–Holm equation and the K dV equation. J. Phys. A 32:L307–L314zbMATHCrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Jaulent M., Jean C. (1976). The inverse problem for the one-dimensional Schrödinger equation with an energy-dependent potential. I. Ann. Inst. H. Poincar Sect. A (N.S.) 25:105–118MathSciNetzbMATHGoogle Scholar
  24. 24.
    Johnson R.S. (2003). On solutions of the Camassa–Holm equation. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459:1687–1708zbMATHADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Li Y.S., Zhang J.E. (2004). The multiple-soliton solution of the Camassa–Holm equation. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 460:2617–2627zbMATHADSGoogle Scholar
  26. 26.
    Li, Y.S., Zhang, J.E.: Analytical multiple-soliton solution of the Camassa–Holm equation. Preprint 2004, J. Nonlinear Math. Phys. (to appear)Google Scholar
  27. 27.
    Liu S.Q., Zhang Y. (2005). Deformations of semisimple bihamiltonian structures of hydrodynamic type. J. Geom. Phys. 54:427–453zbMATHCrossRefADSMathSciNetGoogle Scholar
  28. 28.
    Martĺnez Alonso L. (1980). Schrödinger spectral problems with energy-dependent potentials as sources of nonlinear Hamiltonian evolution equations. J. Math. Phys. 21:2342–2349CrossRefADSMathSciNetGoogle Scholar
  29. 29.
    Matveev V.B., Salle M.A. (1991). Darboux transformations and solitons In: Springer series in nonlinear dynamics. Springer, Berlin Heildelberg New YorkGoogle Scholar
  30. 30.
    McKean H. (2003). The Liouville correspondence between the Korteweg-de Vries and the Camassa-Holm hierarchies. Dedicated to the memory of Jürgen K. Moser. Comm. Pure Appl. Math. 56:998–1015zbMATHMathSciNetGoogle Scholar
  31. 31.
    McKean H. (2004). Breakdown of the Camassa–Holm equation. Comm. Pure Appl. Math. 57:416–418zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Schiff J. (1998). The Camassa–Holm equation: a loop group approach. Physica D 121(1–2):24–43zbMATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingPeople’s Republic of China

Personalised recommendations