Vect(S1) Action on Pseudodifferential Symbols on S1 and (Noncommutative) Hydrodynamic Type Systems
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The standard embedding of the Lie algebra V ect(S1 ) of smooth vector fields on the circle V ect(S1 ) into the Lie algebra D(S1 ) of pseudodifferential symbols on S1 identifies vector field f(x) x V ect(S1 ) and its dual as (f(x) x ) = f(x) (u(x)dx2 ) = u(x)-2 . The space of symbols can be viewed as the space of functions on T S1 . The natural lift of the action of Diff(S1 ) yields Diff(S1 )-module. In this paper we demonstate this construction to yield several examples of dispersionless integrable systems. Using Ovsienko and Roger method for nontrivial deformation of the standard embedding of V ect(S1 ) into D(S1 ) we obtain the celebrated HunteSaxton equation. Finally, we study the Moyal quantization of all such systems to construct noncommutative systems. Dedicated to Professor Dieter Mayer on his 60th birthday
- © 2006, the Authors. Published by Atlantis Press.
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TY - JOUR AU - Partha Guha PY - 2006 DA - 2006/11/01 TI - Vect(S1) Action on Pseudodifferential Symbols on S1 and (Noncommutative) Hydrodynamic Type Systems JO - Journal of Nonlinear Mathematical Physics SP - 549 EP - 565 VL - 13 IS - 4 SN - 1776-0852 UR - https://doi.org/10.2991/jnmp.2006.13.4.9 DO - 10.2991/jnmp.2006.13.4.9 ID - Guha2006 ER -