Euler equations on homogeneous spaces and Virasoro orbits

By B. Khesin, G. Misiołek. 2002

253 citations recorded by the source corpus.

Abstract 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.

Subfields: Geometry and Topology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f0af2fba24bacaad20e6375441d9ce6858678fcc

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Citations in the corpus

Geometry of Diffeomorphism Groups, Complete integrability and Geometric statistics (2013, background): “The proof repeats with minor changes the proof for the case of a metric that is degenerate along a subgroup H; see [16].”
Classical solutions of the periodic Camassa—Holm equation (2002, methodology): “The problem was considered subsequently by Holm, Kouranbaeva, Marsden, Ratiu and Shkoller [HoKMRS] and, independently, in an unpublished paper by A. Constantin and the author (see also a remark in [Sh] and an outline in [ HM1 ]) but using techniques developed primarily to handle the case of integer ”
On a Camassa-Holm type equation with two dependent variables (2005, background): “As it has been long known, V admits a very natural geometrical interpretation of dual of the Witt algebra V of vector fields in ‘one dimension’2, whose elements will be represented with f ∂ ∂x , endowed with the natural Lie bracket.”
Well-posedness of modified Camassa-Holm equations (2009, methodology): “Local well-posedness for (1.2) was discussed by Constantin [6], Constantin and Escher [7] for the initial data in Hs(S), S = [0,2π ] with s 4 and s 3 respectively, and by Misiołek [22] with s > 3/2.”
On Hamiltonian Perturbations of Hyperbolic Systems of Conservation Laws, II: Universality of Critical Behaviour (2005, methodology): “[19]) after doing the following Miura-type transformation u = v − ǫ(2)vxx.”
Analyticity of the Cauchy problem for an integrable evolution equation (2003, result): “All of these manifestations of so-called complete integrability were already studied in [CH] and Camassa, Holm and Hyman [CHH] but other approaches and further results can be found in [FF], Beals, Sattinger and Szmigielski [BSS1], [BSS2], Constantin and McKean [CM] and Khesin and Misiooek [ KM ].”
Low-Regularity Solutions of the Periodic Camassa–Holm Equation (2007, methodology): “Theorem 1 and Periodic Peakons To prove Theorem 1 we follow an approach which has been used previously by several authors, in particular in connection with relating (1) with the geodesic flow on the diffeomorphism group of or on the Bott–Virasoro group (cf. Constantin, 2000; Constantin and Kolev, 20”
On geodesic exponential maps of the Virasoro group (2007, background): “The cases k = 0, 1 were considered in [22, 24, 30] in a somewhat formal way using a purely algebraic approach (cf.”