On integrability of the Camassa-Holm equation and its invariants. A geometrical approach

and was intensively explored afterwards (see, for example, Refs. [5,6,7,17]). Its superizations were also constructed, see [1,19]. Since (1) is not an evolution equation, its integrability properties (existence and even definition of Hamiltonian structures, conservation laws, etc.) are not standard to establish. One of the ways widely used to overcome this difficulty is to introduce a new unknown m = uu xx and transform Eq. (1) to the system

which has almost evolutionary form. We stress this “almost”, because the second equation in (2) (that can be considered as a constrain to the first one) disrupts the picture and, at best, necessitates to invert the operator 1 -D 2 x . At worst, dealing with Eq. ( 2) as with an evolution equation may lead to fallacious results.

In our approach based on the geometrical framework exposed in Ref. [3], we treat the equation at hand as a submanifold in the manifold of infinite jets and consider two natural extensions of this equation, cf. with Ref [16]. The first one is called the ℓ-covering and serves the role of the tangent bundle. The second extension, ℓ * -covering, is the counterpart to the cotangent bundle. The key property of these extensions is that the spaces of their nonlocal (in the sense of [15]) symmetries and cosymmetries contain all essential integrability invariants of the initial equation. The efficiency of the method was tested for a number of problems (see Refs. [12,13,14]) and we apply it to the Camassa-Holm equation here.

In Section 2 we briefly expose the necessary definition and facts. Section 3 contains computations for the Camassa-Holm equation in its matrix version (computations and results are more compact in this representation), while in Section 4 we reformulate them for the original form (1) and compare later the results obtained for the two alternative presentations. Finally, Section 5 contains discussion of the results obtained. Throughout our exposition we use a very stimulating conceptual parallel between categories of smooth manifolds and differential equations proposed initially by A.M. Vinogradov and in its modern form presented in Table 1. This table is not just a toy dictionary but a quite helpful tool to formulate important definitions and results. For example, a bivector on a smooth manifold M may be understood as a derivation of the ring C ∞ (M) with values in C ∞ (T * M). Translating this statement to the language of differential equations we come to the definition of variational bivectors and their description as shadows of symmetries in the ℓ * -covering (see Theorem 2 below). Another example: any vector field (differential 1-form) on M may be treated as a function on T * M (on T M). Hence, to any symmetry (cosymmetry) there corresponds a conservation law on the space of the ℓ * -covering (ℓ-covering). This leads to the notions of nonlocal vectors and forms that, in turn, provide a basis to construct weakly nonlocal structures (see Subsections 3.4 and 3.5). Of course, these parallels are not completely straightforward (in technical aspects, especially), but extremely enlightening and fruitful.

The idea of this paper arose in the discussions one of the authors had with Volodya Roubtsov in 2007. We agreed to write two parallel texts on integrability of the Camassa-Holm equation that reflect our viewpoints. The reader can now compare our results with the ones presented in [20].

We present here a concise exposition of the theoretical background used in the subsequent sections, see Refs. [3,13,15].

Let π : E → M be a fiber bundle and π ∞ : J ∞ (π) → M be the bundle of its infinite jets. To simplify our exposition we shall assume that π is a vector bundle. In all applications below π is the trivial bundle R m × R n → R n . We consider infinite prolongations of differential equations as submanifolds E ⊂ J ∞ (π) and retain the notation π ∞ for the restriction π ∞ | E . Any such a manifold is endowed with the Cartan distribution which spans at every point tangent spaces to the graphs of jets. A symmetry of E is a vector field that preserves this distribution. The set of symmetries is a Lie algebra over R denoted by sym E .

For any equation E its linearization operator ℓ E : κ → P is defined, where κ is the module 1 of sections of the pullback π ∞ (π) and P is the module of sections of some vector bundle over E . Then sym E can be identified with solutions of the equation

For two symmetries ϕ 1 , ϕ 2 ∈ sym E their commutator is denoted by {ϕ 1 , ϕ 2 }.

Denote by Λ i h the module of horizontal i-forms on E and introduce the notation

for any module Q. The adjoint to ℓ E operator

arises and solutions of the equation

are called cosymmetries of E ; the space of cosymmetries is denoted by cosym E . Let

. To any conservation law there corresponds its generating function δ ω ∈ cosym E , where δ :

is the differential in the E 1 term of 1 All the modules below are modules over

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