We consider trigonometric solutions of the Kadomtsev–Petviashvili (KP) hierarchy. It is known that their poles move as particles of the Calogero–Moser model with trigonometric potential. We show that this correspondence can be extended to the level of hierarchies: the evolution of the poles with respect to the kth hierarchical time of the KP hierarchy is governed by a Hamiltonian that is a linear combination of the first k higher Hamiltonians of the trigonometric Calogero–Moser hierarchy.
REFERENCES
1.
H.
Airault
, H. P.
McKean
, and J.
Moser
, “Rational and elliptic solutions of the Korteweg-De Vries equation and a related many-body problem
,” Commun. Pure Appl. Math.
30
, 95
–148
(1977
).2.
F.
Calogero
, “Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials
,” J. Math. Phys.
12
, 419
(1971
).3.
F.
Calogero
, “Exactly solvable one-dimensional many-body systems
,” Lett. Nuovo Cimento
13
, 411
–415
(1975
).4.
J.
Moser
, “Three integrable Hamiltonian systems connected with isospectral deformations
,” Adv. Math.
16
, 197
–220
(1975
).5.
M. A.
Olshanetsky
and A. M.
Perelomov
, “Classical integrable finite-dimensional systems related to Lie algebras
,” Phys. Rep.
71
, 313
–400
(1981
).6.
I. M.
Krichever
, “Rational solutions of the Kadomtsev-Petviashvili equation and integrable systems of N particles on a line
,” Funct. Anal. Appl.
12
(1
), 59
–61
(1978
).7.
D. V.
Chudnovsky
and G. V.
Chudnovsky
, “Pole expansions of non-linear partial differential equations
,” Il Nuovo Cimento B
40
, 339
–350
(1977
).8.
I. M.
Krichever
, “Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles
,” Funk. Anal. i Ego Pril. 14
(4
), 45
–54
(1980
) (in Russian) [Funct. Anal. Appl.
14(4), 282–290 (1980)].9.
T.
Shiota
, “Calogero-Moser hierarchy and KP hierarchy
,” J. Math. Phys.
35
, 5844
–5849
(1994
).10.
L.
Haine
, “KP trigonometric solitons and an adelic flag manifold
,” SIGMA
3
, 015
(2007
).11.
E.
Date
, M.
Jimbo
, M.
Kashiwara
, and T.
Miwa
, Transformation Groups for Soliton Equations: Nonlinear Integrable Systems – Classical Theory and Quantum Theory, Kyoto, 1981
(World Scientific
, Singapore
, 1983
), pp. 39
–119
.12.
M.
Jimbo
and T.
Miwa
, “Solitons and infinite dimensional Lie algebras
,” Publ. Res. Inst. Math. Sci. Kyoto
19
, 943
–1001
(1983
).13.
K.
Takasaki
and T.
Takebe
, “Integrable hierarchies and dispersionless limit
,” Rev. Math. Phys.
07
, 743
–808
(1995
).14.
M.
Adler
and P.
van Moerbeke
, “A matrix integral solution to two-dimensional Wp-gravity
,” Commun. Math. Phys.
147
, 25
–56
(1992
).15.
K.
Takasaki
, “Differential Fay identities and auxiliary linear problem of integrable hierarchies
,” in Exploring New Structures and Natural Constructions in Mathematical Physics
, Advanced Studies in Pure Mathematics, edited by K.
Hasegawa
, T.
Hayashi
, S.
Hosono
, and Y.
Yamada
(Mathematical Society of Japan
, Tokyo
, 2011
), pp. 387
−441
; arXiv:0710.5356.16.
S. M.
Natanzon
and A. V.
Zabrodin
, “Formal solutions to the KP hierarchy
,” J. Phys. A: Math. Theor.
49
, 145206
(2016
).17.
A.
Perelomov
, Integrable Systems of Classical Mechanics and Lie Algebras
(Birkhauser Verlag Basel
, 1990
).18.
S.
Wojciechowski
, “The analogue of the Bäcklund transformation for integrable many-body systems
,” J. Phys. A: Math. Gen.
15
, L653
–L657
(1982
).19.
A. G.
Abanov
, E.
Bettelheim
, and P.
Wiegmann
, “Integrable hydrodynamics of Calogero-Sutherland model: Bidirectional Benjamin-Ono equation
,” J. Phys. A: Math. Theor.
42
, 135201
(2009
).20.
Yu.
Suris
, The Problem of Integrable Discretization: Hamiltonian Approach
(Springer Basel AG
, 2003
).21.
A.
Kasman
and M.
Gekhtman
, “Solitons and almost-intertwining matrices
,” J. Math. Phys.
42
, 3540
–3551
(2001
).22.
A.
Zabrodin
, “Elliptic solutions to integrable nonlinear equations and many-body systems
,” J. Geom. Phys.
146
, 103506
(2019
).© 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.
