We construct the local and nonlocal conserved densities for the Camassa–Holm equation by solving a suitable Riccati equation. We also define a Kadomtsev–Petviashvili extension for the local Camassa–Holm hierarchy.
REFERENCES
1.
Camassa
R.
, and Holm
, D. D.
, “An integrable shallow water equation with peaked solitons
,” Phys. Rev. Lett.
71
, 1661
–1664
(1993
).2.
Falqui
, G.
, Magri
, F.
, and Pedroni
, M.
, “Bihamiltonian geometry, Darboux coverings and linearization of the KP hierarchy
,” Commun. Math. Phys.
197
, 303
–324
(1998
).3.
Hunter
, J.
, and Saxton
, R.
, “Dynamics of director fields
,” SIAM J. Appl. Math.
51
, 1498
–1521
(1991
).4.
Khesin
, B.
and Misiolek
, G.
, “Euler equations on homogeneous spaces and Virasoro orbits
,” Adv. Math.
176
, 116
–144
(2003
).5.
Kruskal
, M.
, “Nonlinear wave equations
,” in Dynamical Systems, Theory and Applications
, edited by J.
Moser
, Lecture Notes in Physics 38
(Springer
, Heidelberg, 1975
), pp. 310
–354
.6.
Lenells
, J.
, “Conservation laws of the Camassa–Holm equation
,” J. Phys. A
38
, 869
–880
(2005
).7.
Lorenzoni
, P.
, and Pedroni
, M.
, “On the bi—Hamiltonian structures of the Camassa–Holm and Harry Dym equations
,” Int. Math. Res. Notices
75
, 4019
–4029
(2004
).8.
Magri
, F.
, Casati
, P.
, Falqui
, G.
, and Pedroni
, M.
, “Eight lectures on integrable systems
,” in Integrability of Nonlinear Systems
, 2nd ed., edited by Y.
Kosmann-Schwarzbach
et al., Lecture Notes in Physics 638
(Springer
, New York, 2004
), pp. 209
–250
.9.
H.
McKean
, “The Liouville correspondence between the Korteweg-de Vries and the Camassa–Holm hierarchies
,” Commun. Pure Appl. Math.
56
, 998
–1015
(2003
).10.
Oevel
,W.
, and Carillo
,S.
, “Squared eigenfunction symmetries for soliton equations. I
,” J. Math. Anal. Appl.
217
, 161
–178
(1998
);Oevel
, W.
and Carillo
, S.
, “Squared eigenfunction symmetries for soliton equations. II
,” J. Math. Anal. Appl.
217
, 179
–199
(1988
).11.
Pedroni
, M.
, Sciacca
, V.
, and Zubelli
, J. P.
, “On the bi-Hamiltonian theory for the Harry Dym equation
,” Theor. Math. Phys.
133
, 1583
–1595
(2002
).© 2005 American Institute of Physics.
2005
American Institute of Physics
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