21 research outputs found
Integrable Quartic Potentials and Coupled KdV Equations
We show a surprising connection between known integrable Hamiltonian systems
with quartic potential and the stationary flows of some coupled KdV systems
related to fourth order Lax operators. In particular, we present a connection
between the Hirota-Satsuma coupled KdV system and (a generalisation of) the
integrable case quartic potential. A generalisation of the case
is similarly related to a different (but gauge related) fourth order Lax
operator. We exploit this connection to derive a Lax representation for each of
these integrable systems. In this context a canonical transformation is derived
through a gauge transformation.Comment: LaTex, 11 page
Quantum Super-Integrable Systems as Exactly Solvable Models
We consider some examples of quantum super-integrable systems and the associated nonlinear extensions of Lie algebras. The intimate relationship between super-integrability and exact solvability is illustrated. Eigenfunctions are constructed through the action of the commuting operators. Finite dimensional representations of the quadratic algebras are thus constructed in a way analogous to that of the highest weight representations of Lie algebras
A new construction of recursion operators for systems of the hydrodynamic type
We consider a certain class of two-dimensional systems of the hydrodynamic type that contains all examples known to us and can be associated with a class of linear wave equations possessing an algebra of ladder operators. We use this to give a simple construction of recursion operators for these systems, not always coinciding with those of Sheftel and Teshukov
Symplectic Maps from Cluster Algebras
We consider nonlinear recurrences generated from the iteration of maps that arise from cluster algebras. More precisely, starting from a skew-symmetric integer matrix, or its corresponding quiver, one can define a set of mutation operations, as well as a set of associated cluster mutations that are applied to a set of affine coordinates (the cluster variables). Fordy and Marsh recently provided a complete classification of all such quivers that have a certain periodicity property under sequences of mutations. This periodicity implies that a suitable sequence of cluster mutations is precisely equivalent to iteration of a nonlinear recurrence relation. Here we explain briefly how to introduce a symplectic structure in this setting, which is preserved by a corresponding birational map (possibly on a space of lower dimension). We give examples of both integrable and non-integrable maps that arise from this construction. We use algebraic entropy as an approach to classifying integrable cases. The degrees of the iterates satisfy a tropical version of the map
On non-QRT Mappings of the Plane
We construct 9-parameter and 13-parameter dynamical systems of the plane
which map bi-quadratic curves to other bi-quadratic curves and return to the
original curve after two iterations. These generalize the QRT maps which map
each such curve to itself. The new families of maps include those that were
found as reductions of integrable lattices
Differential constraints compatible with linearized equations
Differential constraints compatible with the linearized equations of partial
differential equations are examined. Recursion operators are obtained by
integrating the differential constraints
Discrete integrable systems and Poisson algebras from cluster maps
We consider nonlinear recurrences generated from cluster mutations applied to
quivers that have the property of being cluster mutation-periodic with period
1. Such quivers were completely classified by Fordy and Marsh, who
characterised them in terms of the skew-symmetric matrix that defines the
quiver. The associated nonlinear recurrences are equivalent to birational maps,
and we explain how these maps can be endowed with an invariant Poisson bracket
and/or presymplectic structure.
Upon applying the algebraic entropy test, we are led to a series of
conjectures which imply that the entropy of the cluster maps can be determined
from their tropical analogues, which leads to a sharp classification result.
Only four special families of these maps should have zero entropy. These
families are examined in detail, with many explicit examples given, and we show
how they lead to discrete dynamics that is integrable in the Liouville-Arnold
sense.Comment: 49 pages, 3 figures. Reduced to satisfy journal page restrictions.
Sections 2.4, 4.5, 6.3, 7 and 8 removed. All other results remain, with minor
editin
Extensions of the matrix Gelfand-Dickey hierarchy from generalized Drinfeld-Sokolov reduction
The matrix version of the -KdV hierarchy has been recently
treated as the reduced system arising in a Drinfeld-Sokolov type Hamiltonian
symmetry reduction applied to a Poisson submanifold in the dual of the Lie
algebra . Here a
series of extensions of this matrix Gelfand-Dickey system is derived by means
of a generalized Drinfeld-Sokolov reduction defined for the Lie algebra
using the natural
embedding for any positive integer. The
hierarchies obtained admit a description in terms of a matrix
pseudo-differential operator comprising an -KdV type positive part and a
non-trivial negative part. This system has been investigated previously in the
case as a constrained KP system. In this paper the previous results are
considerably extended and a systematic study is presented on the basis of the
Drinfeld-Sokolov approach that has the advantage that it leads to local Poisson
brackets and makes clear the conformal (-algebra) structures related to
the KdV type hierarchies. Discrete reductions and modified versions of the
extended -KdV hierarchies are also discussed.Comment: 60 pages, plain TE
Quantized W-algebra of sl(2,1) and quantum parafermions of U_q(sl(2))
In this paper, we establish the connection between the quantized W-algebra of
and quantum parafermions of that a
shifted product of the two quantum parafermions of
generates the quantized W-algebra of
Cluster mutation-periodic quivers and associated Laurent sequences
We consider quivers/skew-symmetric matrices under the action of mutation (in
the cluster algebra sense). We classify those which are isomorphic to their own
mutation via a cycle permuting all the vertices, and give families of quivers
which have higher periodicity. The periodicity means that sequences given by
recurrence relations arise in a natural way from the associated cluster
algebras. We present a number of interesting new families of non-linear
recurrences, necessarily with the Laurent property, of both the real line and
the plane, containing integrable maps as special cases. In particular, we show
that some of these recurrences can be linearised and, with certain initial
conditions, give integer sequences which contain all solutions of some
particular Pell equations. We extend our construction to include recurrences
with parameters, giving an explanation of some observations made by Gale.
Finally, we point out a connection between quivers which arise in our
classification and those arising in the context of quiver gauge theories.Comment: The final publication is available at www.springerlink.com. 42 pages,
35 figure