Abstract
In a recent paper we proved that for certain class of perturbations of the hyperbolic equation u t = f (u)u x , there exist changes of coordinate, called quasi-Miura transformations, that reduce the perturbed equations to the unperturbed one. We prove in the present paper that if in addition the perturbed equations possess Hamiltonian structures of certain type, the same quasi-Miura transformations also reduce the Hamiltonian structures to their leading terms. By applying this result, we obtain a criterion of the existence of Hamiltonian structures for a class of scalar evolutionary PDEs and an algorithm to find out the Hamiltonian structures.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Camassa R. and Holm D.D. (1993). An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71: 1661–1664
Camassa R., Holm D.D. and Hyman J.M. (1994). A new integrable shallow water equation. Adv. Appl. Mech. 31: 1–33
Degasperis, A., Procesi, M.: Asymptotic integrability. In: Symmetry and Perturbation Theory (Rome, 1998), pp. 23–37, World Scientific Publishing, River Edge (1999)
Degasperis, A., Holm, D.D., Hone, A.N.I.: A new integrable equation with peakon solutions. Teoret. Mat. Fiz. 133, 170–183 (2002). Translation in Theoret. Math. Phys. 133, 1463–1474 (2002)
Degiovanni L., Magri F. and Sciacca V. (2005). On deformation of Poisson manifolds of hydrodynamic type. Commun. Math. Phys. 253: 1–24
Drinfeld, V., Sokolov, V.: Lie algebras and equations of Korteweg-de Vries type, J. Math. Sci. 30, 1975–2036 (1985). Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki (Noveishie Dostizheniya) 24, 81–180 (1984)
Dubrovin B. (2006). On Hamiltonian perturbations of hyperbolic systems of conservation laws. II. Universality of critical behaviour. Commun. Math. Phys. 267: 117–139
Dubrovin B., Liu S.-Q. and Zhang Y. (2006). On Hamiltonian perturbations of hyperbolic systems of conservation laws, I: Quasi-triviality of bi-Hamiltonian perturbations. Commun. Pure Appl. Math. 59: 559–615
Dubrovin, B., Zhang, Y.: Normal forms of integrable PDEs, Frobenius manifolds and Gromov–Witten invariants. Preprint arXiv: math.DG/0108160 (2001)
Flato M., Lichnerowicz A. and Sternheimer D. (1976). Deformations of Poisson brackets, Dirac brackets and applications. J. Math. Phys. 17: 1754–1762
Flato M., Pinczon G. and Simon J. (1977). Nonlinear representations of Lie groups. Ann. Sci. École Norm. Sup. (4) 10: 405–418
Flato, M., Simon, J., Taflin, E.: Asymptotic completeness, global existence and the infrared problem for the Maxwell–Dirac equations. Mem. Am. Math. Soc. 127(606) (1997)
Fuchssteiner B. and Fokas A.S. (1981). Symplectic structures, their Bäcklund transformations and hereditary symmetries. Physica D 4: 47–66
Getzler E. (2002). A Darboux theorem for Hamiltonian operators in the formal calculus of variations. Duke Math. J. 111: 535–560
Hone A. and Wang J. (2003). Prolongation algebras and Hamiltonian operators for peakon equations. Inverse Probl 19: 129–145
Kaup D. (1980). On the inverse scattering problem for cubic eigenvalue problems of the class \(\psi_{xxx}+6Q\psi_{x}+6R\psi=\lambda\psi\). Stud. Appl. Math. 62: 189–216
Lichnerowicz A. (1977). Les varietes de Poisson et leurs algèbres de Lie associeés. J. Diff. Geom. 12: 253–300
Liu S.-Q. and Zhang Y. (2005). Deformations of semisimple bihamiltonian structures of hydrodynamic type. J. Geom. Phys. 54: 427–453
Liu S.-Q. and Zhang Y. (2006). On quasi-triviality and integrability of a class of scalar evolutionary PDEs. J. Geom. Phys. 57: 101–119
Sawada K. and Kotera T. (1974). A method for finding N-soliton solutions of the KdV equation and KdV-like equation. Progr. Theor. Phys. 51: 1355–1367
Taflin E. (1983). Analytic linearization of the Korteweg-de Vries equation. Pac. J. Math. 108: 203–220
Taflin E. (1981). Analytic linearization, Hamiltonian formalism and infinite sequences of constants of motion for the Burgers equation. Phys. Rev. Lett. 47: 1425–1428
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, SQ., Wu, CZ. & Zhang, Y. On Properties of Hamiltonian Structures for a Class of Evolutionary PDEs. Lett Math Phys 84, 47–63 (2008). https://doi.org/10.1007/s11005-008-0234-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11005-008-0234-y