Remarks on the Conserved Densities of the Camassa-holm Equation

It is pointed out that the higher-order symmetries of the Camassa-Holm (CH) equation are nonlocal and nonlocality poses problems to obtain higher-order conserved densities for this integrable equation (J. Phys. A: Math. Gen. 2005, 38 869-880). This difficulty is circumvented by defining a nolinear hierarchy for the CH equation and an explicit expression is constructed for the nth-order conserved density. The Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations are quasi-linear in the sense that the dispersive behaviour of the solution of each equation is governed by a linear term giving the order of the equation. The dispersion produced is compensated by nonlinear effects resulting in the formation of exponentially localized solitons. In early 1990's, Rosenau and Hyman [1] introduced a family of nonlinear evolution equations given by K(p, q) : u t + (u q) x + (u p) 3x = 0 , 1 < q = p ≤ 3. (1) For the restriction imposed on p, the dispersive term of (1) is nonlinear. We shall call evolution equations with nonlinear dispersive terms as fully nonlinear evolution (FNE) equations. The solitary wave solutions of (1) were found to have compact support. That is, they vanish identically outside the finite range. These solutions were given the name compactons. For special values of q and p one can have peaked and cusp-like solutions often called peakon and cuspon [2]. Besides being nonintegrable, another awkward analytical constraint for the equations in (1) is that they do not follow from an action principle. Almost simultaneously with the work of Rosenau and Hyman, Camassa and Holm [3] introduced another FNE equation given by u t − u xxt + 3uu x − 2u x u 2x − uu 3x = 0 .