Geodesic flow on the diffeomorphism group of the circle

By Adrian Constantin, Boris Kolev. 2003

384 citations recorded by the source corpus.

Subfields: Applied Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/229809362922ed5ce7440aadc03688331463783b

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In Applied Mathematics

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Infinite propagation speed for a two component Camassa-Holm equation (2009, background): “It is interesting to note that for the CH equation these diffeomorphisms have a geometric interpretation in the context of geodesic flow [ 19 , 18], however currently no such interpretation exists for this two-component generalisation.”
Global Existence and Blow-Up Phenomena for the Degasperis-Procesi Equation (2006, methodology): “Furthermore, the Camassa-Holm equation is a re-expression of geodesic flow on the diffeomorphism group [ 13 ] or on the Bott-Virasoro group [40], while no geometric derivation of the Degasperis-Procesi equation is available.”
Inverse scattering transform for the Camassa–Holm equation (2006, background): “CH also arises as an equation of the geodesic flow for the H 1 right-invariant metrics on the Bott–Virasoro group (if ω > 0) and on the diffeomorphism group (if ω = 0) [8, 12, 13, 34].”
Some Geometric Evolution Equations Arising as Geodesic Equations on Groups of Diffeomorphisms Including the Hamiltonian Approach (2006, methodology): “But there is a mistake in [13], p 795, where the authors assume that composition and inversion on H(S(1)) are smooth.”
Compactly Supported Solutions of the Camassa-Holm Equation (2005, background): “4) is not accidental: it represents the conservation of momentum in the physical variables (see [10, 11]).”
On the Blow-up Phenomena for the Degasperis–Procesi Equation (2010, methodology): “Both the KdV equation [20] and the Camassa-Holm equation [1, 6, 8, 13, 14] are completely integrable models for the propagation of shallow water waves.”
On the Regularity of the Composition of Diffeomorphisms (2011, methodology): “In particular, Riemann exponential maps have been studied in [10], [11], [20], [21].”
WELL-POSEDNESS OF HIGHER-ORDER CAMASSA-HOLM EQUATIONS (2009, methodology): “1) reads in the cases k = 0, 1, 2, 3 as follows [14].”
Traveling wave solutions of the Degasperis-Procesi equation (2005, background): “3) is a re-expression of geodesic flow [4,7,18] whereas no such geo interpretation is valid for (1.”
Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group (2011, background): “and for s > 1 they are related to the higher order Camassa Holm equations, see [5].”