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1.
We use the Jacobi method to construct various integrable systems, such as the Stäckel systems and Toda chains, related to various root systems. We find canonical transformations that relate integrals of motion for the generalized open Toda chains of types B
n, C
n, and D
n. 相似文献
2.
We introduce a criterion that a given bi-Hamiltonian structure admits a local coordinate system where both brackets have
constant coefficients. This criterion is applied to the bi-Hamiltonian open Toda lattice in a generic point, which is shown
to be locally isomorphic to a Kronecker odd-dimensional pair of brackets with constant coefficients. This shows that the open
Toda lattice cannot be locally represented as a product of two bi-Hamiltonian structures. Near, a generic point, the bi-Hamiltonian
periodic Toda lattice is shown to be isomorphic to a product of two open Toda lattices (one of which is a (trivial) structure
of dimension 1). While the above results might be obtained by more traditional methods, we use an approach based on general
results on geometry of webs. This demonstrates the possibility of applying a geometric language to problems on bi-Hamiltonian
integrable systems; such a possibility may be no less important than the particular results proved in this paper. Based on
these geometric approaches, we conjecture that decompositions similar to the decomposition of the periodic Toda lattice exist
in local geometry of the Volterra system, the complete Toda lattice, the multidimensional Euler top, and a regular bi-Hamiltonian
Lie coalgebra. We also state general conjectures about the geometry of more general "homogeneous" finite-dimensional bi-Hamiltonian
structures. The class of homogeneous structures is shown to coincide with the class of systems integrable by Lenard scheme.
The bi-Hamiltonian structures which admit a non-degenerate Lax structure are shown to be locally isomorphic to the open Toda
lattice. 相似文献
3.
We use p-component fermions, p = 2, 3,..., to represent (2p−2)N-fold integrals as a fermionic vacuum expectation. This yields
a fermionic representation for various (2p−2)-matrix models. We discuss links with the p-component Kadomtsev-Petviashvili
hierarchy and also with the p-component Toda lattice hierarchy. We show that the set of all but two flows of the p-component
Toda lattice hierarchy changes standard matrix models to new ones.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 265–277, August, 2007. 相似文献
4.
Plamen Iliev 《Selecta Mathematica, New Series》2008,13(3):497-530
We consider the heat equation u
t
= Lu where L is a second-order difference operator in a discrete variable n. The fundamental solution has an expansion in terms of the Bessel functions of imaginary argument. The coefficients α
k
(n, m) in this expansion are analogs of Hadamard’s coefficients for the (continuous) Schr?dinger operator.
We derive an explicit formula for α
k
in terms of the wave and the adjoint wave functions of the Toda lattice hierarchy. As a first application of this result,
we prove that the values of these coefficients on the diagonals n = m and n = m + 1 define a hierarchy of differential-difference equations which is equivalent to the Toda lattice hierarchy. Using this
fact and the correspondence between commutative rings of difference operators and algebraic curves we show that the fundamental
solution can be summed up, giving a finite formula involving only two Bessel functions with polynomial coefficients in the
time variable t, if and only if the operator L belongs to the family of bispectral operators constructed in [18]. 相似文献
5.
Plamen Iliev 《Selecta Mathematica, New Series》2007,13(3):497-530
We consider the heat equation u
t
= Lu where L is a second-order difference operator in a discrete variable n. The fundamental solution has an expansion in terms of the Bessel functions of imaginary argument. The coefficients α
k
(n, m) in this expansion are analogs of Hadamard’s coefficients for the (continuous) Schr?dinger operator.
We derive an explicit formula for α
k
in terms of the wave and the adjoint wave functions of the Toda lattice hierarchy. As a first application of this result,
we prove that the values of these coefficients on the diagonals n = m and n = m + 1 define a hierarchy of differential-difference equations which is equivalent to the Toda lattice hierarchy. Using this
fact and the correspondence between commutative rings of difference operators and algebraic curves we show that the fundamental
solution can be summed up, giving a finite formula involving only two Bessel functions with polynomial coefficients in the
time variable t, if and only if the operator L belongs to the family of bispectral operators constructed in [18].
相似文献
6.
We discuss a one-to-one correspondence between the polynomial first integrals of Hamiltonian systems with exponential interaction and the hyperintegrals of the two-dimensional Toda lattice. We establish formulas for recalculating the corresponding polynomials and some general properties of their algebraic structure. 相似文献
7.
Masahiko Ito 《Compositio Mathematica》2001,129(3):325-340
We state certain product formulae for Jackson integrals associated with irreducible reduced root systems. The Jackson integral is defined here as a sum over any full-rank sublattice of the coweight lattice for the root system. In particular, a Weyl group symmetry classification of the Jackson integrals is done when they have an expression of a product of the Jacobi elliptic theta functions. Most of the product formulae investigated by Aomoto, Macdonald and Gustafson appear in the list of classifications. A new product formula for an F
4 root system is included in it. 相似文献
8.
A. Yu. Orlov 《Theoretical and Mathematical Physics》2006,146(2):183-206
It is known that resonant multisoliton solutions depend on higher times and a set of parameters (integrals of motion). We
show that soliton tau functions of the Toda lattice (and of the multicomponent Toda lattice) are tau functions of a dual hierarchy,
where the higher times and the parameters (integrals of motion) exchange roles. The multisoliton solutions turn out to be
rational solutions of the dual hierarchy, and the infinite-soliton tau functions turn out to be hypergeometric-type tau functions
of the dual hierarchy. The variables in the dual hierarchies exchange roles. Soliton momenta are related to the Frobenius
coordinates of partitions in the decomposition of rational solutions with respect to Schur functions. As an example, we consider
partition functions of matrix models: their perturbation series is, on one hand, a hypergeometric tau function and, on the
other hand, can be interpreted as an infinite-soliton solution.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 2, pp. 222–250, February, 2006. 相似文献
9.
Consider the two-dimensional Toda lattice, with certain skew-symmetric initial condition, which is preserved along the locus
of the space of time variables. Restricting the solution to , we obtain another hierarchy called Pfaff lattice, which has its own tau function, being equal to the square root of the
restriction of 2D-Toda tau function. We study its bilinear and Fay identities, W and Virasoro symmetries, relation to symmetric and symplectic matrix integrals and quasiperiodic solutions.
Received: 20 September 1999 / Published online: 1 February 2002 相似文献
10.
Geng Xianguo 《数学学报(英文版)》1992,8(1):78-90
Under the constraint determined by a relation (a n ,b n )T={f(?)} n between the reflectionless potentials and the eigenfunctions of the general discrete Schrödinger eigenvalue problem, the Lax pair of the Toda lattice is nonlinearized to be a finite-dimensional difference system and a nonlinear evolution equation, while the solution varietyN of the former is an invariant set of S-flows determined by the latter, and the constants of the motion for the algebraic system are presented.f maps the solution of the algebraic system into the solution of a certain stationary Toda equation. Similar results concerning the Langmuir lattice are given, and a relation between the two difference systems, which are the spatial parts of the nonlinearized Lax pairs of the Toda lattice and Langmuir lattice, is discussed. 相似文献
11.
Hiraku Abe 《Differential Geometry and its Applications》2013,31(5):577-593
A completely integrable system on a symplectic manifold is called super-integrable when the number of independent integrals of motion is more than half the dimension of the manifold. Several important completely integrable systems are super-integrable: the harmonic oscillators, the Kepler system, the non-periodic Toda lattice, etc. Motivated by an additional property of the super-integrable system of the Toda lattice (Agrotis et al., 2006) [2], we will give a generalization of the Atiyah and Guillemin–Sternberg?s convexity theorem. 相似文献
12.
The notion of Laplace invariants is generalized to lattices and discrete equations that are difference analogues of hyperbolic
partial differential equations with two independent variables. The sequence of Laplace invariants satisfies the discrete analogue
of the two-dimensional Toda lattice. We prove that terminating this sequence by zeros is a necessary condition for the existence
of integrals of the equation under consideration. We present formulas for the higher symmetries of equations possessing such
integrals. We give examples of difference analogues of the Liouville equation.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 2, pp. 271–284, November, 1999. 相似文献
13.
S. V. Kryukov 《Theoretical and Mathematical Physics》1995,105(2):1359-1368
Higher-order local integrals of motion for the classical sine-Gordon system are calculated. For the kth integral of motion, the general formula is obtained. It is discussed how the integrals of motion can be used for constructing classical lattice Wn-algebras. A direct consideration is given for the case of W2.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 2, pp. 214–224, November, 1995. 相似文献
14.
Augustin-Liviu Mare 《Advances in Mathematics》2004,185(2):347-369
Consider the infinite-dimensional flag manifold LK/T corresponding to the simple Lie group K of rank l and with maximal torus T. We show that, for K of type A, B or C, if we endow the space (where q1,…,ql+1 are multiplicative variables) with an -bilinear product satisfying some simple properties analogous to the quantum product on QH∗(K/T), then the isomorphism type of the resulting ring is determined by the integrals of motion of a certain periodic Toda lattice system, in exactly the same way as the isomorphism type of QH∗(K/T) is determined by the integrals of motion of the non-periodic Toda lattice (see (Ann. Math. 149 (1999) 129)). This is an infinite-dimensional extension of the main result of Mare (Relations in the quantum cohomology ring of G/B, preprint math. DG/0210026) and at the same time a generalization of M.A. Guest and T. Otofuji (Comm. Math. Phys. 217 (2001) 475). 相似文献
15.
We consider a two-parameter process Xz defined by the sum of multiple Skorohod integrals and ordinary Lebesgue integrals. A generalized Ito's formula is given. We also introduce a two-parameter analog of the SkorohodStratonovich integral and establish an Ito's formula in the Stratonovich form 相似文献
16.
We show that Toda lattices with the Cartan matrices A
n
, B
n
, C
n
, and D
n
are Liouville-type systems. For these systems of equations, we obtain explicit formulas for the invariants and generalized Laplace invariants. We show how they can be used to construct conservation laws (x and y integrals) and higher symmetries. 相似文献
17.
We consider hyperbolic systems of equations that have full sets of integrals along both characteristics. The best known example
of models of this type is given by two-dimensional open Toda chains. For systems that have integrals, we construct a differential
operator that takes integrals into symmetries. For systems of the chosen type, this proves the existence of higher symmetries
dependent on arbitrary functions.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 2, pp. 344–355, May, 2008. 相似文献
18.
A. Okounkov 《Selecta Mathematica, New Series》2001,7(1):57-81
We use representation theory to obtain a number of exact results for random partitions. In particular, we prove a simple determinantal formula for correlation functions of what we call the Schur measure on partitions (which is a far reaching generalization of the Plancherel measure; see [3], [8]) and also observe that these correlations functions are t \tau -functions for the Toda lattice hierarchy. We also give a new proof of the formula due to Bloch and the author [5] for the so-called n-point functions of the uniform measure on partitions and comment on the local structure of a typical partition. 相似文献
19.
Generalizing the graded commutator in superalgebras, we propose a new bracket operation on the space of graded operators with an involution. We study properties of this operation and show that the Lax representation of the two-dimensional N=(1|1) supersymmetric Toda lattice hierarchy can be realized via the generalized bracket operation; this is important in constructing the semiclassical (continuum) limit of this hierarchy. We construct the continuum limit of the N=(1|1) Toda lattice hierarchy, the dispersionless N=(1|1) Toda hierarchy. In this limit, we obtain the Lax representation, with the generalized graded bracket becoming the corresponding Poisson bracket on the graded phase superspace. We find bosonic symmetries of the dispersionless N=(1|1) supersymmetric Toda equation. 相似文献
20.
We introduce a renormalization procedure for the τ-function of integrable systems. We illustrate the procedure using the supercritical Toda shock problem as a model problem. We start with a finite chain and take the limit of the solution as the number of particles N → ∞. This results in a new formula for the τ-function for the problem with an infinite chain. We apply the renormalized formula to rederive leading-order effects of the supercritical Toda shock problem. © 1998 John Wiley & Sons, Inc. 相似文献