Bi-Hamiltonian structure of a dynamical system introduced by Braden and Hone

We investigate the finite dimensional dynamical system derived by Braden and Hone in 1996 from the solitons of A n−1 affine Toda field theory. This system of evolution equations for an Hermitian matrix L and a real diagonal matrix q with distinct eigenvalues was interpreted as a special case of the spin Ruijsenaars–Schneider models due to Krichever and Zabrodin. A decade later, Li re-derived the model from a general framework built on coboundary dynamical Poisson groupoids. This led to a Hamiltonian description of the gauge invariant content of the model, where the gauge transformations act as conjugations of L by diagonal unitary matrices. Here, we point out that the same dynamics can be interpreted also as a special case of the spin Sutherland systems obtained by reducing the free geodesic motion on symmetric spaces, studied by Pusztai and the author in 2006; the relevant symmetric space being . This construction provides an alternative Hamiltonian interpretation of the Braden–Hone dynamics. We prove that the two Poisson brackets are compatible and yield a bi-Hamiltonian description of the standard commuting flows of the model.

Note:

18 pages, references and some explanations added in v2

dynamical system
Poisson bracket
invariance: gauge
transformation: gauge
field theory: Toda
evolution equation
geodesic
soliton

References(29)

Figures(0)

DOI:
- 10.1093/imrn/rnp130

Affine Toda solitons and systems of Calogero-Moser type

H.W. Braden
(
- Edinburgh U.
)
,
Andrew N.W. Hone
(
- Edinburgh U.
)

- Phys.Lett.B 380 (1996) 296-302
•
e-Print:
- hep-th/9603178
•
DOI:
- 10.1016/0370-2693(96)00499-6

On the Hamiltonian formulation of the trigonometric spin Ruijsenaars–Schneider system

Oleg Chalykh
(
- Leeds U., Math.
)
,
Maxime Fairon
(
- Glasgow U.
)

- Lett.Math.Phys. 110 (2020) 11, 2893-2940,
- Lett.Math.Phys. (2020),
- Lett.Math.Phys. 110 (2020) 2893-2940
•
e-Print:
- 1811.08727
•
DOI:
- 10.1007/s11005-020-01320-x

DOI:
- 10.1007/s002200050292

Spin versions of the complex trigonometric Ruijsenaars-Schneider model from cyclic quivers

Maxime Fairon

- J.Integrab.Syst. 4 (2019) 1, xyz008
•
e-Print:
- 1811.08717
•
DOI:
- 10.1093/integr/xyz008

BiHamiltonian geometry, Darboux coverings, and linearization of the KP hierarchy

Gregorio Falqui
(
- SISSA, Trieste
)
,
Franco Magri
(
- Milan U. and
- Poincare Inst.
)
,
Marco Pedroni
(
- Genoa U. and
- Poincare Inst.
)

- Commun.Math.Phys. 197 (1998) 303-324
•
e-Print:
- solv-int/9806002
•
DOI:
- 10.1007/s002200050452

DOI:
- 10.1016/j.geomphys.2016.04.023

Poisson-Lie analogues of spin Sutherland models

L. Feher
(
- Szeged U. and
- Wigner RCP, Budapest
)

- Nucl.Phys.B 949 (2019) 114807
•
e-Print:
- 1809.01529
•
DOI:
- 10.1016/j.nuclphysb.2019.114807

On the duality between the hyperbolic Sutherland and the rational Ruijsenaars-Schneider models

- J.Phys.A 42 (2009) 185202
•
e-Print:
- 0901.1983
•
DOI:
- 10.1088/1751-8113/42/18/185202

Spin Calogero models obtained from dynamical r-matrices and geodesic motion

L. Feher
(
- Budapest, RMKI and
- Szeged U.
)
,
B.G. Pusztai
(
- Montreal U., CRM and
- Concordia U., Math. Dept.
)

- Nucl.Phys.B 734 (2006) 304-325
•
e-Print:
- math-ph/0507062
•
DOI:
- 10.1016/j.nuclphysb.2005.11.025

Spin Calogero models associated with Riemannian symmetric spaces of negative curvature

L. Feher
(
- Budapest, RMKI and
- Szeged U.
)
,
B.G. Pusztai
(
- Montreal U., CRM and
- Concordia U., Math. Dept.
)

- Nucl.Phys.B 751 (2006) 436-458
•
e-Print:
- math-ph/0604073
•
DOI:
- 10.1016/j.nuclphysb.2006.06.029

DOI:
- 10.1016/0167-2789(84)90015-0

Canonical spectral coordinates for the Calogero-Moser space associated with the cyclic quiver

- J.Nonlin.Math.Phys. 27 (2020) 2, 243-266,
- J. Nonlinear Math. Phy. 27 (2020) 243-266
•
e-Print:
- 1812.02544
•
DOI:
- 10.1080/14029251.2020.1700634