## Abstract

This paper is a survey article on bi-Hamiltonian systems on the dual of the Lie algebra of vector fields on the circle. Here, we investigate the special case where one of the structures is the canonical Lie–Poisson structure and the second one is constant. These structures, called affine or modified Lie–Poisson structures, are involved in the integrability of certain Euler equations that arise as models for shallow water waves.

## Footnotes

One contribution of 13 to a Theme Issue ‘Water waves’.

↵In this case, the group is just the rotation group,

*SO*(3).↵However, this formalism seems to have been extended to hydrodynamics before Arnold by Moreau (1959).

↵The affine structure on the Virasoro algebra, which makes the KdV equation a bi-Hamiltonian system, seems to have been first discovered by Gardner (1971) and for this reason, some authors call it the

*Gardner bracket,*see also Faddeev & Zakharov (1971).↵The expression ‘Hamiltonian manifold’ is often used for the generalization of Poisson structure in the case of infinite-dimension manifolds.

↵The Schouten–Nijenhuis bracket is an extension of the Lie bracket of vector fields to skew-symmetric multivector fields (see Vaisman 1994).

↵This means that the corresponding Hamiltonian vector fields

*X*_{f1}, …,*X*_{fn}are independent on an open dense subset of*M*.↵A first integral is a function which is constant on the trajectories of the vector field.

↵This terminology is used for the evolution equations in infinite dimension.

↵Here, d

_{m}*f*, the differential of a function*f*∈*C*^{∞}() at*m*∈, is to be understood as an element of the Lie algebra .↵In what follows, the convention for lower or upper indices may be confusing since we shall deal with tensors on both and . Therefore, we emphasize that the convention we use in this paper is the following: upper indices correspond to contravariant tensors on and therefore covariant tensors on , whereas lower indices correspond to covariant tensors on and therefore contravariant tensors on .

↵A Poisson structure on a linear space is

*affine*if the bracket of two linear functionals is an affine functional.↵This corresponds to the opposite of the usual Lie bracket of vector fields.

↵In the sequel, we use the notation

*u*,*v*, … for elements of and*m*,*n*, … for elements of to distinguish them, although they all belong to*C*^{∞}(*S*^{1}).↵The coadjoint action of a Lie algebra on its dual is defined aswhere

*u*,*v*∈,*m*∈, and the pairing is the standard one between and .↵The second order geodesic equation corresponding to a one-sided invariant metric on a Lie group can always be reduced to a first-order quadratic equation on the dual of the Lie algebra of the group: the Euler equation (see Arnold & Khesin (1998) or Kolev (2004)). The generality of this reduction was first revealed by Poincaré (1901) and applied to hydrodynamics by Arnold (1966).

↵Note that our

*m*corresponds to*u*in the notations of Lax (1976).↵The proof of lemma 4.3 can be found in Olver (1993), while the proof of lemma 4.4 can be found in Lax (1976).

↵Using a theorem of Peetre (1959), a local cochain can be characterized by the condition

↵Recall that ∂

*m*is the linear differential operator defined by- © 2007 The Royal Society

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