The Camassa Holm equation: conserved quantities and the initial value problem

Abstract

Using a Miura-Gardner-Kruskal type construction, we show that the Camassa-Holm equation has an infinite number of local conserved quantities. We explore the implications of these conserved quantities for global well-posedness.

Geometric Integrability of the Camassa–Holm Equation. II

“…This is exactly what happens in the case of the CH and ACH equations: Schiff observed in [49] that the ACH equation (41) admits a loop group interpretation, and that this point of view can be used to derive Bácklund transformations for ACH. Furthermore, M. Fisher and Schiff showed in 1999 (see [19]) that it is possible to compute conservation laws of (41) by means of the classical Miura-Gardner-Kruskal method [32]. On the other hand, as explained in Section 1, a direct construction of conservation laws for CH was carried out only in 2002 (see [43]) and a Bácklund-like transformation for CH has appeared only recently (see [26] and Section 4 of the present paper).…”

“…D Thus, if the "infinitesimal deformation" of the variables if along X is measured by G, and the corresponding deformation of the nonlocal variables y b is measured by the functions H b (i.e., \£ t = G a and y b = H b ), then heuristically Equations (18) and (19) tell us that X is a nonlocal 7r-symmetry of the system E a = 0 if and only if the linearized equations…”

“…Note that transformation (40) is nonlocal: the new independent variable y is a "potential variable" which can be determined from x, t, p, and u only through integration in the (x, t) plañe (see [8] for information about these "potential" transformations). In particular, it is not an invertible transformation, as explained in [19,49], and so CH and ACH are not equivalent equations. Transformations such as (40) can change essentially the character of the nonlinearity of the initial equation (see [37,51]) so that properties that are not apparent for it may be successfully studied for the transformed equation.…”

“…We remark that this construction was the first one which successfully generated an infinite number of nontrivial local conservation laws for the CH equation. Other constructions are in the original papers by Camassa and Holm [10,11] (in which they genérate explicitly three local conservation laws of CH using the bi-hamiltonian formalism), in [19] (in which conservation laws for CH are generated via hodographic transformations from conservation laws for the associated Camassa-Holm (ACH) equation introduced by Schiff in [49]), and in the very interesting work [33] (in which the bi-hamiltonian approach is effectively shown to genérate an infinite number of local conservation laws).…”

Pseudo-potentials, nonlocal symmetries and integrability of some shallow water equations

“…As in the KdV case, one can use (72) and (74) to construct conservation laws for the CH and HS equations [10,15,20,37,38]. Setting γ = ∞ n=1 γ n λ n/2 yields the conserved densities…”