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The Camassa-Holm Hierarchy, N-Dimensional Integrable Systems, and Algebro-Geometric Solution on a Symplectic Submanifold

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Abstract

This paper shows that the Camassa-Holm (CH) spectral problem yields two different integrable hierarchies of nonlinear evolution equations (NLEEs), one is of negative order CH hierachy while the other one is of positive order CH hierarchy. The two CH hierarchies possess the zero curvature representations through solving a key matrix equation. We see that the well-known CH equation is included in the negative order CH hierarchy while the Dym type equation is included in the positive order CH hierarchy. Furthermore, under two constraint conditions between the potentials and the eigenfunctions, the CH spectral problem is cast in: 1. a new Neumann-like N-dimensional system when it is restricted into a symplectic submanifold of ℝ2N which is proven to be integrable by using the Dirac-Poisson bracket and the r-matrix process; and 2. a new Bargmann-like N-dimensional system when it is considered in the whole ℝ2N which is proven to be integrable by using the standard Poisson bracket and the r-matrix process.

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References

  1. Ablowitz, M.J., Segur, H.: Soliton and the Inverse Scattering Transform. Philadelphia: SIAM, 1981

  2. Alber, M.S., Roberto Camassa, Fedorov, Y.N., Holm, D.D., Marsden, J.E.: The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE's of shallow water and Dym type. Commun. Math. Phys. 221, 197–227 (2001)

    MathSciNet  MATH  Google Scholar 

  3. Alber, M.S., Roberto Camassa, Holm, D.D., Marsden, J.E.: The geometry of peaked solitons and billiard solutions of a class of integrable PDEs. Lett. Math. Phys. 32, 137–151 (1994)

    MathSciNet  MATH  Google Scholar 

  4. Alber, M.S., Fedorov, Y.N.: Wave solution of evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians. J. Phys. A: Math. Gen. 33, 8409–8425 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alber, M.S., Fedorov, Y.N.: Algebraic geometrical solutions for certain evolution equations and Hamiltonian flows on nonlinear subvarieties of generalized Jacobians. Inv. Prob. 17 (2001), to appear

  6. Arnol'd, V.I.: Mathematical Methods of Classical Mechanics. Berlin: Springer-Verlag, 1978

  7. Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Calogero, F.: An integrable Hamiltonian system. Phys. Lett. A 201, 306–310 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cao, C.W.: Nonlinearization of Lax system for the AKNS hierarchy. Sci. China A (in Chinese) 32, 701–707 (1989); also see English Edition: Nonlinearization of Lax system for the AKNS hierarchy. Sci. Sin. A 33, 528–536 (1990)

    Google Scholar 

  10. Dickey, L.A.: Soliton Equations and Hamiltonian Systems. Singapore: World Scientific, 1991

  11. Dubrovin, B.: Riemann surfaces and nonlinear equations. I. Moscow: MGU, 1986

  12. Dubrovin, B.: Theta-functions and nonlinear equations. Russ. Math. Surv. 36, 11–92 (1981)

    MATH  Google Scholar 

  13. Faddeev, L.D., Takhtajan, L.A.: Hamiltonian Methods in the Theory of Solitons. Berlin: Springer-Verlag, 1987

  14. Gesztesy, F., Holden, H.: Algebraic-geometric solutions of the Camassa-Holm hierarchy. Private communication, to appear in Revista Mat. Iberoamericana

  15. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. New York: Wiley, 1978

  16. Newell, A.C.: Soliton in Mathematical Physics. Philadelphia: SIAM, 1985

  17. Novikov, S.P., Manakov, S.V., Pitaevskii, L.P., Zakharov, V.E.: Theory of Solitons. The Inverse Scattering Method. New York: Plenum, 1984

  18. Qiao, Z.J., Cao, C.W., Strampp, W.F.: Category of nonlinear evolution equations, algebraic structure, and r-matrix. J. Math. Phys. 44, 701–722 (2003)

    Article  Google Scholar 

  19. Qiao, Z.J.: Generalized r-matrix structure and algebro-geometric solutions for integrable systems. Rev. Math. Phys. 13, 545–586 (2001)

    Article  MathSciNet  Google Scholar 

  20. Ragnisco, O., Bruschi, M.: Peakons, r-matrix and Toda Lattice. Physica A 228, 150–159 (1996)

    MathSciNet  Google Scholar 

  21. Sklyanin, E.K.: Separation of variables. Prog. Theor. Phys. Suppl. 118, 35–60 (1995)

    MathSciNet  MATH  Google Scholar 

  22. Suris, Y.B.: A discrete time peakons lattice. Phys. Lett. A 217, 321–329 (1996)

    Article  Google Scholar 

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  1. Los Alamos National Laboratory, Los Alamos, T-7 and CNLS, MS B-284, NM 87545, USA

    Zhijun Qiao

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  1. Zhijun Qiao
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Qiao, Z. The Camassa-Holm Hierarchy, N-Dimensional Integrable Systems, and Algebro-Geometric Solution on a Symplectic Submanifold. Commun. Math. Phys. 239, 309–341 (2003). https://doi.org/10.1007/s00220-003-0880-y

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  • DOI: https://doi.org/10.1007/s00220-003-0880-y

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