We show that the following three systems related to various hydrodynamical approximations: the Korteweg-de Vries equation, the Camassa-Holm equation, and the Hunter-Saxton equation, have the same symmetry group and similar bihamiltonian structures. It turns out that their configuration space is the Virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different right-invariant metrics on this group or on appropriate homogeneous spaces. In particular, we describe how Arnold's approach to the Euler equations as geodesic flows of one-sided invariant metrics extends from Lie groups to homogeneous spaces.We also show that the above three cases describe all generic bihamiltonian systems which are related to the Virasoro group and can be integrated by the translation argument principle: they correspond precisely to the three different types of generic Virasoro orbits. Finally, we discuss interrelation between the above metrics and Kahler structures on Virasoro orbits as well as open questions regarding integrable systems corresponding to a finer classification of the orbits.

Abstract. We study an equation lying 'mid-way' between the periodic HunterSaxton and Camassa-Holm equations, and which describes evolution of rotators in liquid crystals with external magnetic field and self-interaction. We prove that it is an Euler equation on the diffeomorphism group of the circle corresponding to a natural right-invariant Sobolev metric. We show that the equation is bihamiltonian and admits both cusped as well as smooth traveling-wave solutions which are natural candidates for solitons. We also prove that it is locally well-posed and establish results on the lifespan of its solutions. Throughout the paper we argue that despite similarities to the KdV, CH and HS equations, the new equation manifests several distinctive features that set it apart from the other three.

Abstract. We study a family of equations defined on the space of tensor densities of weight λ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct "peakon" and "multi-peakon" solutions for all λ = 0, 1, and "shock-peakons" for λ = 3. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold's approach to Euler equations on Lie groups.

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Abstract. It is shown that a strong solution of the Camassa-Holm equation, initially decaying exponentially together with its spacial derivative, must be identically equal to zero if it also decays exponentially at a later time. In particular, a strong solution of the Cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution.