Integrable systems, multicomponent twisted Heisenberg-Virasoro algebra and its central extensions

Abstract

Let {{\displaystyle \mathcal{D}}}_N be the multicomponent twisted Heisenberg-Virasoro algebra. We compute the second continuous cohomology group with coeficients in ℂ and study the bihamiltonian Euler equations associated to {{\displaystyle \mathcal{D}}}_N and its central extensions.

Keywords

Bihamiltonian Euler equation / multicomponent twisted Heisenberg-Virasoro algebra

References

[1] Antonowicz M, Fordy A P. Factorisation of energy dependent Schrödinger operators: Miura maps and modified systems. Comm Math Phys, 1989, 124(3): 465–486

[2] Arbarello E, Concini C, Kac V G, Procesi C. Moduli spaces of curves and representation theory. Comm Math Phys, 1988, 117(1): 1–36

[3] Arnold V I. Sur la géométrie différentielle des groupes de Lie de dimenson infinie et ses applications à l’hydrodynamique des fluids parfaits. Ann Inst Fourier (Grenoble), 1966, 16: 319–361

[4] Arnold V I, Khesin B A. Topological Methods in Hydrodynamics. Appl Math Sci, Vol 125. New York: Springer-Verlag, 1998

[5] Broer L F. Approximate equations for long water waves. Appl Sci Res, 1975, 31: 377–395

[6] Chen M, Liu S-Q, Zhang Y. Hamiltonian structures and their reciprocal transformations for the r-KdV-CH hierarchy. J Geom Phys, 2009, 59(9): 1227–1243

[7] Ebin D G, Marsden J. Groups of diffeomorphisms and the notion of an incompressible fluid. Ann of Math, 1970, 92: 102–163

[8] Ge Y Y, Zuo D. A new class of Euler equation on the dual of the N= 1 extended Neveu-Schwarz algebra. J Math Phys, 2018, 59(11): 113505 (8 pp)

[9] Guha P, Olver P J. Geodesic flow and two (super) component analog of the Camassa-Holm equation. SIGMA Symmetry Integrability Geom Methods Appl, 2006, 2: Paper 054 (9 pp)

[10] Harnad J, Kupershmidt B A. Symplectic geometries on T*G, Hamiltonian group actions and integrable systems. J Geom Phys, 1995, 16: 168–206

[11] Holm D D, Ivanov R I. Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples. J Phys A, 2010, 43(49): 492001 (20 pp)

[12] Kaup D J. A higher-order water-wave equation and the method for solving it. Progr Theoret Phys, 1975, 54: 396–408

[13] Khesin B, Misiolek G. Euler equations on homogeneous spaces and Virasoro orbits. Adv Math, 2003, 176: 116–144

[14] Khesin B A, Wendt R. The Geometry of Infinite-Dimensional Groups. Ergeb Math Grenzgeb (3), Vol 51. New York: Springer-Verlag, 2009

[15] Kirillov A A. Orbits of the group of diffeomorphisms of a circle and local superalgebras. Funct Anal Appl, 1980, 15: 135–137

[16] Kirillov A A. Infinite dimensional Lie groups: their orbits, invariants and representations. The geometry of moments. In: Doebner H D, Palev T D, eds. Twistor Geometry and Non-Linear Systems. Lecture Notes in Math, Vol 970. New York: Springer-Verlag, 1982, 101–123

[17] Kolev B. Bihamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations. Philos Trans Roy Soc A, 2007, 365: 2333–2357

[18] Kupershmidt B A. A coupled Korteweg-de Vries equation with dispersion. J Phys A, 1985, 18: L571–L573

[19] Kupershmidt B A. Mathematics of dispersive water waves. Comm Math Phys, 1985, 99: 51–73

[20] Kupershmidt B A. Lie algebras and Korteweg-de Vries equations. Phys D, 1987, 27: 294–310

[21] Marcel P, Ovsienko V, Roger C. Extension of the Virasoro and Neveu-Schwarz algebras and generalized Sturm-Liouville operators. Lett Math Phys, 1997, 40: 31–39

[22] Misiolek G. A shallow water equation as a geodesic flow on the Bott-Virasoro group. J Geom Phys, 1998, 24: 203–208

[23] Ovsienko V, Khesin B. The super Korteweg-de Vries equation as an Euler equation. Funct Anal Appl, 1988, 21: 329–331

[24] Strachan I A B, Szablikowski B. Novikov algebras and a classification of multicomponent Camassa-Holm equations. Stud Appl Math, 2014, 133: 84–117

[25] Whitham G B. Variational methods and applications to water waves. Proc Roy Soc Lond Ser A, 1967, 299: 6–25

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Higher Education Press