163 research outputs found
The Perlick system type I: from the algebra of symmetries to the geometry of the trajectories
In this paper, we investigate the main algebraic properties of the maximally
superintegrable system known as "Perlick system type I". All possible values of
the relevant parameters, and , are considered. In particular,
depending on the sign of the parameter entering in the metrics, the motion
will take place on compact or non compact Riemannian manifolds. To perform our
analysis we follow a classical variant of the so called factorization method.
Accordingly, we derive the full set of constants of motion and construct their
Poisson algebra. As it is expected for maximally superintegrable systems, the
algebraic structure will actually shed light also on the geometric features of
the trajectories, that will be depicted for different values of the initial
data and of the parameters. Especially, the crucial role played by the rational
parameter will be seen "in action".Comment: 16 pages, 7 figure
dbar-approach to the dispersionless KP hierarchy
The dispersionless limit of the scalar nonlocal dbar-problem is derived. It
is given by a special class of nonlinear first-order equations. A
quasi-classical version of the dbar-dressing method is presented. It is shown
that the algebraic formulation of dispersionless hierarchies can be expressed
in terms of properties of Beltrami tupe equations. The universal Whitham
hierarchy and, in particular, the dispersionless KP hierarchy turn out to be
rings of symmetries for the quasi-classical dbar-problem.Comment: 13 pages, LaTex 24.9K
Dynamical R-Matrices for Integrable Maps
The integrability of two symplectic maps, that can be considered as
discrete-time analogs of the Garnier and Neumann systems is established in the
framework of the -matrix approach, starting from their Lax representation.
In contrast with the continuous case, the -matrix for such discrete systems
turns out to be of dynamical type; remarkably, the induced Poisson structure
appears as a linear combination of compatible ``more elementary" Poisson
structures. It is also shown that the Lax matrix naturally leads to define
separation variables, whose discrete and continuous dynamics is investigated.Comment: 16 plain tex page
Non-linear WKB Analysis of the String Equation
We apply non-linear WKB analysis to the study of the string equation. Even
though the solutions obtained with this method are not exact, they approximate
extremely well the true solutions, as we explicitly show using numerical
simulations. ``Physical'' solutions are seen to be separatrices corresponding
to degenerate Riemann surfaces. We obtain an analytic approximation in
excellent agreement with the numerical solution found by Parisi et al. for the
case.Comment: 21 pages. To appear in the proceedings of the Research Conference on
Advanced Quantum Field Theory and Critical Phenomena, held in Como (Italy),
June 17-21, 1991 -- World Scientifi
Backlund transformations and Hamiltonian flows
In this work we show that, under certain conditions, parametric Backlund
transformations (BTs) for a finite dimensional integrable system can be
interpreted as solutions to the equations of motion defined by an associated
non-autonomous Hamiltonian. The two systems share the same constants of motion.
This observation lead to the identification of the Hamiltonian interpolating
the iteration of the discrete map defined by the transformations, that indeed
will be a linear combination of the integrals appearing in the spectral curve
of the Lax matrix. An application to the Toda periodic lattice is given.Comment: 19 pages, 2 figures. to appear in J. Phys.
Classical Dynamical Systems from q-algebras:"cluster" variables and explicit solutions
A general procedure to get the explicit solution of the equations of motion
for N-body classical Hamiltonian systems equipped with coalgebra symmetry is
introduced by defining a set of appropriate collective variables which are
based on the iterations of the coproduct map on the generators of the algebra.
In this way several examples of N-body dynamical systems obtained from
q-Poisson algebras are explicitly solved: the q-deformed version of the sl(2)
Calogero-Gaudin system (q-CG), a q-Poincare' Gaudin system and a system of
Ruijsenaars type arising from the same (non co-boundary) q-deformation of the
(1+1) Poincare' algebra. Also, a unified interpretation of all these systems as
different Poisson-Lie dynamics on the same three dimensional solvable Lie group
is given.Comment: 19 Latex pages, No figure
The δ̅ -approach to the dispersionless KP hierarchy
The dispersionless limit of the scalar nonlocal a-problem is derived. It is given by a special class of nonlinear first-order equations. A quasiclassical version of the partial derivative -dressing method is presented. It is shown that the algebraic formulation of dispersionless hierarchies can be expressed in terms of properties of Beltrami-type equations. The universal Whitham hierarchy and, in particular, the dispersionless KP hierarchy turn out to be rings of symmetries for the quasiclassical partial derivative -problem
A Novel Hierarchy of Integrable Lattices
In the framework of the reduction technique for Poisson-Nijenhuis structures,
we derive a new hierarchy of integrable lattice, whose continuum limit is the
AKNS hierarchy. In contrast with other differential-difference versions of the
AKNS system, our hierarchy is endowed with a canonical Poisson structure and,
moreover, it admits a vector generalisation. We also solve the associated
spectral problem and explicity contruct action-angle variables through the
r-matrix approach.Comment: Latex fil
Backlund transformations for many-body systems related to KdV
We present Backlund transformations (BTs) with parameter for certain
classical integrable n-body systems, namely the many-body generalised
Henon-Heiles, Garnier and Neumann systems. Our construction makes use of the
fact that all these systems may be obtained as particular reductions
(stationary or restricted flows) of the KdV hierarchy; alternatively they may
be considered as examples of the reduced sl(2) Gaudin magnet. The BTs provide
exact time-discretizations of the original (continuous) systems, preserving the
Lax matrix and hence all integrals of motion, and satisfy the spectrality
property with respect to the Backlund parameter.Comment: LaTeX2e, 8 page
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