Abstract
We consider a new partial differential equation recently obtained by Degasperis and Procesi using the method of asymptotic integrability; this equation has a form similar to the Camassa–Holm shallow water wave equation. We prove the exact integrability of the new equation by constructing its Lax pair and explain its relation to a negative flow in the Kaup–Kupershmidt hierarchy via a reciprocal transformation. The infinite sequence of conserved quantities is derived together with a proposed bi-Hamiltonian structure. The equation admits exact solutions as a superposition of multipeakons, and we describe the integrable finite-dimensional peakon dynamics and compare it with the analogous results for Camassa–Holm peakons.
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REFERENCES
R. Camassa and D. D. Holm, Phys. Rev. Lett., 71, 1661–1664 (1993).
R. Camassa, D. D. Holm, and J. M. Hyman, Adv. Appl. Mech., 31, 1–33 (1994).
B. Fuchssteiner, Phys. D, 95, 229–243 (1996).
H. Dullin, G. Gottwald, and D. D. Holm, Phys. Rev. Lett., 87, 1945–1948 (2001).
A. N. W. Hone, Appl. Math. Lett., 13, 37–42 (2000); R. A. Kraenkel and A. Zenchuk, Phys. Lett. A, 260, 218–224 (1999).
A. Degasperis and M. Procesi, “Asymptotic integrability,” in: Symmetry and Perturbation Theory (A. Degasperis and G. Gaeta, eds.), World Scientific, Singapore (1999), pp. 23–37.
C. Gilson and A. Pickering, J. Phys. A, 28, 2871–2888 (1995).
A. Ramani, B. Dorizzi, and B. Grammaticos, Phys. Rev. Lett., 49, 1538–1541 (1982).
A. N. W. Hone, J. Phys. A, 32, L307–314 (1999).
A. Degasperis, A. N. W. Hone, and D. D. Holm, “A class of equations with peakon and pulson solutions,” (in preparation).
P. R. Gordoa and A. Pickering, J. Math. Phys., 40, 5749–5786 (1999).
J. G. Kingston and C. Rogers, Phys. Lett. A, 92, 261–264 (1982); C. Rogers, “Reciprocal transformations and their applications,” in: Nonlinear Evolutions (J. Leon, ed.), World Scientific, Singapore (1988), pp. 109–123; C. Rogers, “Bäcklund transformations in soliton theory,” in: Soliton Theory: A Survey of Results (A. P. Fordy, ed.), Manchester Univ. Press, Manchester (1990), pp. 97-130.
A. N. W. Hone, Phys. Lett. A, 263, 347–354 (1999).
J. Weiss, M. Tabor, and G. J. Carnevale, J. Math. Phys., 24, 522–526 (1983).
G. Tzitzeica, C. R. Acad. Sci. Paris, 150, 955–956, 1227–1229 (1910).
R. K. Dodd and R. K. Bullough, Proc. Roy. Soc. London A, 352, 481–502 (1977).
P. Olver and P. Rosenau, Phys. Rev. E, 53, 1900–1906 (1996); I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, Wiley, New York (1993).
O. Fringer and D. D. Holm, Phys. D, 150, 237–263 (2001).
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Degasperis, A., Holm, D.D. & Hone, A.N.W. A New Integrable Equation with Peakon Solutions. Theoretical and Mathematical Physics 133, 1463–1474 (2002). https://doi.org/10.1023/A:1021186408422
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DOI: https://doi.org/10.1023/A:1021186408422