共查询到20条相似文献,搜索用时 31 毫秒
1.
The bi-Hamiltonian structure of integrable supersymmetric extensions of the Korteweg-de Vries (KdV) equation related to theN=1 and theN=2 superconformal algebras is found. It turns out that some of these extensions admit inverse Hamiltonian formulations in terms of presymplectic operators rather than in terms of Poisson tensors. For one extension related to theN=2 case additional symmtries are found with bosonic parts that cannot be reduced to symmetries of the classical KdV. They can be explained by a factorization of the corresponding Lax operator. All the bi-Hamiltonian formulations are derived in a systematic way from the Lax operators. 相似文献
2.
Any classical r-matrix on the Lie algebra of linear operators on a real vector space V gives rise to a quadratic Poisson structure on V which admits a deformation quantization stemming from the construction of V. Drinfel'd [Dr], [Gr]. We exhibit in this article
an example of quadratic Poisson structure which does not arise this way.
Received: 31 May 2001 / Accepted: 17 August 2001 相似文献
3.
Koji Hasegawa 《Communications in Mathematical Physics》1997,187(2):289-325
For Belavin's elliptic quantum R-matrix, we construct an L-operator as a set of difference operators acting on functions on
the type A weight space. According to the fundamental relation RLL=LLR, taking the trace of the L-operator gives a set of commuting difference operators. We show that for the above mentioned L-operator
this approach gives Macdonald type operators with elliptic theta function coefficient, actually equivalent to Ruijsenaars'
operators. The relationship between the difference L-operator and Krichever's Lax matrix is given, and an explicit formula
for elliptic commuting differential operators is derived. We also study the invariant subspace for the system which is spanned
by symmetric theta functions on the weight space.
Received: 27 December 1995 / Accepted: 11 November 1996 相似文献
4.
L. A. Dickey 《Communications in Mathematical Physics》1981,82(3):361-375
A symplectic structure is constructed and the Liouville integration carried out for a stationary Lax equation [L, P]=0, whereL is a scalar differential operator of an arbitrary order.n
th order operators are included into the variety of first-order matrix operators, and properties of this inclusion are studied. 相似文献
5.
Field equations for generalized principal chiral models with non-constant metric and their possible Lax formulation are considered.
Ansatz for Lax operators is taken linear in currents. Results of a complete investigation of models allowing Lax formulation
with linear ansatz for Lax operators on solvable 2- and 3-dimensional groups are given; all such models appear to be almost
linear. Also models on simple groupSU(2) with diagonal metric are considered; it turns out that Lax formulation exists in this case for constant metrics only.
Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June
2001.
This work was partially supported by grant No. 1929/2001 of Fund for Development of Higher Education. 相似文献
6.
It has been shown by Olshanetsky and Perelomov that the Toda molecule equations associated with any Lie groupG describe special geodesic motions on the Riemannian non-compact symmetric space which is the quotient of the normal real form ofG, G
N, by its maximal compact subgroup. This is explained in more detail and it is shown that the fundamental Poisson bracket relation involving the Lax operatorA and leading to the Yang-Baxter equation and integrability properties is a direct consequence of the fact that the Iwasawa decomposition forG
N endows the symmetric space with a hidden group theoretic structure.Supported by CNP
q
(Brasil) 相似文献
7.
Maarten Bergvelt Michael Gekhtman Alex Kasman 《Mathematical Physics, Analysis and Geometry》2009,12(2):181-200
Pairs of n×n matrices whose commutator differ from the identity by a matrix of rank r are used to construct bispectral differential operators with r×r matrix coefficients satisfying the Lax equations of the Matrix KP hierarchy. Moreover, the bispectral involution on these
operators has dynamical significance for the spin Calogero particles system whose phase space such pairs represent. In the
case r = 1, this reproduces well-known results of Wilson and others from the 1990’s relating (spinless) Calogero-Moser systems to
the bispectrality of (scalar) differential operators.
相似文献
8.
Following Sato's famous construction of the KP hierarchy as a hierarchy of commuting Lax equations on the algebra of microdifferential operators, it is shown that n-reduction leads to a recursive scheme for these equations. Explicit expressions for the recursion operators and the Hamiltonian operators are obtained. 相似文献
9.
L. A. Dickey 《Letters in Mathematical Physics》1995,35(3):229-236
10.
We consider odd Laplace operators acting on densities of various weights on an odd Poisson (= Schouten) manifold M. We prove that the case of densities of weight 1/2 (half-densities) is distinguished by the existence of a unique odd Laplace operator depending only on a point of an 'orbit space' of volume forms. This includes earlier results for the odd symplectic case, where there is a canonical odd Laplacian on half-densities. The space of volume forms on M is partitioned into orbits by the action of a natural groupoid whose arrows correspond to the solutions of the quantum Batalin–Vilkovisky equations. We compare this situation with that of Riemannian and even Poisson manifolds. In particular, we show that the square of an odd Laplace operator is a Poisson vector field defining an analog of Weinstein's 'modular class'. 相似文献
11.
We introduce a class of spin Calogero-Moser systems associated with root systems of simple Lie algebras and give the associated
Lax representations (with spectral parameter) and fundamental Poisson bracket relations. The associated integrable models
(called integrable spin Calogero-Moser systems in the paper) and their Lax pairs are then obtained via Poisson reduction and
gauge transformations. For Lie algebras of A
n
-type, this new class of integrable systems includes the usual Calogero-Moser systems as subsystems. Our method is guided
by a general framework which we develop here using dynamical Lie algebroids.
Received: 19 October 2001 / Accepted: 7 June 2002 Published online: 21 October 2002
RID="*"
ID="*" Research partially supported by NSF grant DMS00-72171. 相似文献
12.
Ian Marshall Michael Semenov-Tian-Shansky 《Communications in Mathematical Physics》2008,284(2):537-552
We combine the projective geometry approach to Schroedinger equations on the circle and differential Galois theory with the
theory of Poisson Lie groups to construct a natural Poisson structure on the space of wave functions (at the zero energy level).
Applications to KdV-like nonlinear equations are discussed. The same approach is applied to 2nd order difference operators on a one-dimensional lattice, yielding an extension of the lattice Poisson Virasoro algebra. 相似文献
13.
A chain of one-dimensional Schrödinger operators connected by successive Darboux transformations is called the ``Darboux chain' or ``dressing chain'. The periodic dressing chain with period $N$ has a control parameter $\alpha$. If $\alpha \not= 0$, the $N$-periodic dressing chain may be thought of as a generalization of the fourth or fifth (depending on the parity of $N$) Painlevé equations . The $N$-periodic dressing chain has two different Lax representations due to Adler and to Noumi and Yamada. Adler's $2 \times 2$ Lax pair can be used to construct a transition matrix around the periodic lattice. One can thereby define an associated ``spectral curve' and a set of Darboux coordinates called ``spectral Darboux coordinates'. The equations of motion of the dressing chain can be converted to a Hamiltonian system in these Darboux coordinates. The symplectic structure of this Hamiltonian formalism turns out to be consistent with a Poisson structure previously studied by Veselov, Shabat, Noumi and Yamada. 相似文献
14.
Luen-Chau Li 《Communications in Mathematical Physics》1999,203(3):573-592
Given a classical r-matrix on a Poisson algebra, we show how to construct a natural family of compatible Poisson structures for the Hamiltonian
formulation of Lax equations. Examples for which our formalism applies include the Benny hierarchy, the dispersionless Toda
lattice hierarchy, the dispersionless KP and modified KP hierarchies, the dispersionless Dym hierarchy, etc.
Received: 10 February 1998 / Accepted: 9 December 1998 相似文献
15.
Pantelis A. Damianou 《Reports on Mathematical Physics》1997,40(3):443-453
With the exception of some minor results and some conjectures, this paper is a survey of the finite nonperiodic Toda lattices and some of their generalizations. The areas investigated include Lax pairs, master symmetries, recursion operators, higher Poisson brackets, invariants, and group symmetries for such systems. 相似文献
16.
CHEN Jin-Bing GENG Xian-Guo 《理论物理通讯》2005,44(9)
This paper deals with the integrability of a finite-dimensional Hamiltonian system linked with the generalized coupled KdV hierarchy. For this purpose the associated Lax representation is presented after an elementary calculation. It is shown that the Lax representation enjoys a dynamical r-matrix formula instead of a classical one in the Poisson bracket on R2N. Consequently the resulting system is proved to be completely integrable in view of its r-matrix structure. 相似文献
17.
CHEN Jin-Bing GENG Xian-Guo 《理论物理通讯》2005,44(3):393-395
This paper deals with the integrability of a finite-dimensional Hamiltonian system linked with the generalized coupled KdV hierarchy. For this purpose the associated Lax representation is presented after an elementary calculation. It is shown that the Lax representation enjoys a dynamical r-matrix formula instead of a classical one in the Poisson bracket on R^2N. Consequently the resulting system is proved to be completely integrable in view of its r-matrix structure. 相似文献
18.
Using the Poisson current algebra of the supersymmetric principal chiral model, we develop the algebraic canonical structure
of the model by evaluating the fundamental Poisson bracket of the Lax matrices that fits into the r–s matrix formalism of
non-ultralocal integrable models. The fundamental Poisson bracket has been used to compute the Poisson bracket algebra of
the monodromy matrix that gives the conserved quantities in involution.
PACS 11.30.Pb; 02.30.Ik 相似文献
19.
Theodore Voronov 《Communications in Mathematical Physics》1999,205(2):315-336
We consider the following construction of quantization. For a Riemannian manifold $M$ the space of forms on T
⋆
M
is made into a space of (full) symbols of operators acting on forms on M. This gives rise to the composition of symbols, which is a deformation of the (“super”)commutative multiplication of forms.
The symbol calculus is exact for differential operators and the symbols that are polynomial in momenta. We calculate the symbols
of natural Laplacians. (Some nice Weitzenb?ck like identities appear here.) Formulae for the traces corresponding to natural
gradings of Ω (T
⋆
M
) are established. Using these formulae, we give a simple direct proof of the Gauss–Bonnet–Chern Theorem. We discuss these
results in connection with a general question of the quantization of forms on a Poisson manifold.
Received: 12 November 1998 / Accepted: 1 March 1999 相似文献
20.
The structure of the commutant of Laplace operators in the enveloping and Poisson algebra of certain generalized ax +b groups leads (in this article) to a determination of classical and quantum mechanical first integrals to generalized periodic and non-periodic Toda lattices. Certain new Hamiltonian systems of Toda lattice type are also shown to fit in this framework. Finite dimensional Lax forms for the (periodic) Toda lattices are given generalizing results of Flaschke.Research partially supported by NSF grant MCS 79-03223Research partially supported by NSF grant MCS 79-03153 相似文献