A two-parameters family of Bäcklund transformations for the classical elliptic Gaudin model is constructed. The maps are explicit, symplectic, preserve the same integrals as for the continuous flows, and are a time discretization of each of these flows. The transformations can map real variables into real variables, sending physical solutions of the equations of motion into physical solutions. The starting point of the analysis is the integrability structure of the model. It is shown how the analogue transformations for the rational and trigonometric Gaudin model are a limiting case of this one. An application to a particular case of the Clebsch system is given.

1.
Amico
,
L.
,
Falci
,
G.
, and
Fazio
,
R.
, “
The BCS model and the off-shell Bethe ansatz for vertex models
,”
J. Phys. A
34
,
6425
6434
(
2001
).
2.
Armitage
,
J. V.
and
Eberlein
,
W. F.
,
Elliptic Functions
,
London Mathematical Society Student Texts
Vol. 67 (
Cambridge University Press
,
Cambridge, England
,
2006
).
3.
Belavin
,
A. A.
and
Drinfel'd
,
V. G.
, “
Solutions of the classical Yang-Baxter equation for simple Lie-algebras
,”
Funktsional. Anal. i Prilozhen.
16
,
1
29
(
1982
).
4.
Clebsch
,
A.
, “
Über die Bewegung eines Körpers in einer Flüssigkeit
,”
Math. Ann.
3
,
238
262
(
1870
).
5.
Dean
,
D. J.
and
Hjorth-Jensen
,
M.
, “
Pairing in nuclear systems: From neutron stars to finite nuclei
,”
Rev. Mod. Phys.
75
,
73
(
2003
).
6.
Faddeev
,
L. D.
and
Takhtajan
,
L. A.,
Hamiltonian Methods in the Theory of Solitons
(
Springer-Verlag
,
Berlin
,
1987
).
7.
Gaudin
,
M.
, “
Diagonalisation d'une classe d'hamiltoniens de spin
,”
Le Journal de Physique
37
(
10
),
1087
1098
(
1976
).
8.
Hone
,
A. N.
,
Kuznetsov
,
V. B.
, and
Ragnisco
,
O.
, “
Bäcklund transformations for the sl(2) Gaudin magnet
,”
J. Phys. A
34
,
2477
2490
(
2001
).
9.
Jurčo
,
B.
, “
Classical Yang-Baxter equations and quantum integrable systems
,”
J. Math. Phys.
30
,
1289
1293
(
1989
).
10.
Kalnins
,
E. G.
,
Kuznetsov
,
V. B.
, and
Miller
 Jr,
W.
, “
Quadrics on complex Riemannian spaces of constant curvature, separation of variables and the Gaudin magnet
,”
J. Math. Phys.
35
,
1710
1731
(
1994
).
11.
Kirchhoff
,
G.
,
Vorlesungen über mathematische Physik, Neunzenthe Vorlesung
(
Teubner
,
Leipzig
,
1876
), pp.
233
250
.
12.
Komarov
,
I. V.
, “
Integrable system connected with the Coulomb three-body problem near two-particles thresholds
,”
J. Phys. A
21
,
1191
1197
(
1988
).
13.
Kuznetsov
,
V. B.
and
Sklyanin
,
E. K.
, “
On Bäcklund transformations for many-body systems
,”
J. Phys. A
31
,
2241
2251
(
1998
).
14.
Kuznetsov
,
V. B.
and
Vanhaecke
,
P.
, “
Bäcklund transformations for finite-dimensional integrable systems: A geometric approach
,”
J. Geom. Phys.
806
,
1
40
(
2002
).
15.
Milne-Thomson
,
L. M.
,
Theoretical Hydrodynamics
(
Dover
,
New York
,
1996
).
16.
Musso
,
F.
,
Petrera
,
M.
, and
Ragnisco
,
O.
, “
Algebraic extension of Gaudin models
,”
J. Nonlinear Math. Phys.
12
(
1
),
482
498
(
2005
).
17.
Petrera
,
M.
and
Ragnisco
,
O.
, “
From
$\mathfrak {su}$
su
(2) Gaudin models to integrable tops
,”
Sigma
3
,
058
(
2007
).
18.
Petrera
,
M.
and
Suris
,
Y. B.
, “
An integrable discretization of the rational
$\mathfrak {su}$
su
(2) Gaudin model and related systems
,”
Commun. Math. Phys.
283
,
227
253
(
2008
).
19.
Ragnisco
,
O.
and
Zullo
,
F.
, “
Bäcklund transformations for the trigonometric Gaudin magnet
,”
Sigma
6
,
012
(
2010
).
20.
Ragnisco
,
O.
and
Zullo
,
F.
, “
Bäcklund transformations as exact integrable time-discretizations for the trigonometric Gaudin model
,”
J. Phys. A
43
,
434029
(
2010
).
21.
Ragnisco
,
O.
and
Zullo
,
F.
, “
Bäcklund transformations for the Kirchhoff top
,”
Sigma
7
,
001
(
2011
).
22.
Reyman
,
A. G.
and
Semenov-Tian-Shansky
,
M. A.
, “
Group theoretical methods in the theory of finite-dimensional integrable systems
,” in
Encyclopedia of Mathematical Sciences
,
Dynamical systems VII
(
Springer
,
Berlin
,
1994
), Vol. 16.
23.
Sklyanin
,
E. K.
, “
Separation of variables in Gaudin model
,”
J. Sov. Math.
47
,
2473
2488
(
1989
).
24.
Sklyanin
,
E. K.
, “
Separation of variables: New trends
,”
Prog. Theor. Phys. Suppl.
118
,
35
60
(
1995
).
25.
Sklyanin
,
E. K.
, “
Canonicity of Bäcklund transformations: r-Matrix approach II
,”
Proc. Steklov Inst. Math.
226
,
121
126
(
1999
);
Sklyanin
,
E. K.
, translation from
Tr. Mat. Inst. Steklova
226
,
134
139
(
1999
).
26.
Sklyanin
,
E. K.
, “
Bäcklund transformation and Baxter's Q-operator
,” in
Integrable Systems, From Classical to Quantum
, edited by
J.
Harnad
,
G.
Sabidussi
, and
P.
Winternitz
(
American Mathematical Society
,
Providence
,
2000
), pp.
227
250
.
27.
Sklyanin
,
E. K.
, “
Canonicity of Bäcklund transformation: r-Matrix approach I
,”
L. D. Faddeev's Seminar on Mathematical Physics
,
American Mathmetical Society Translations Series 2
Vol. 201 (
American Mathematical Society
,
Providence
,
2000
), pp.
277
282
.
28.
Sklyanin
,
E. K.
and
Takebe
,
T.
, “
Algebraic Bethe ansatz for the XYZ Gaudin model
,”
Phys. Lett. A
219
,
217
225
(
1996
).
29.
Sklyanin
,
E. K.
and
Takebe
,
T.
, “
Separation of variables in the elliptic Gaudin model
,”
Commun. Math. Phys.
204
,
17
38
(
1998
).
You do not currently have access to this content.