149 research outputs found
A family of linearizable recurrences with the Laurent property
We consider a family of non-linear recurrences with the Laurent property. Although these recurrences are not generated by mutations in a cluster algebra, they fit within the broader framework of Laurent phenomenon algebras, as introduced recently by Lam and Pylyavskyy. Furthermore, each member of this family is shown to be linearizable in two different ways, in the sense that its iterates satisfy both a linear relation with constant coefficients and a linear relation with periodic coefficients. Associated monodromy matrices and first integrals are constructed, and the connection with the dressing chain for Schrödinger operators is also explained
Mutation-Periodic Quivers, Integrable Maps and Associated Poisson Algebras
We consider a class of map, recently derived in the context of cluster
mutation. In this paper we start with a brief review of the quiver context, but
then move onto a discussion of a related Poisson bracket, along with the
Poisson algebra of a special family of functions associated with these maps. A
bi-Hamiltonian structure is derived and used to construct a sequence of Poisson
commuting functions and hence show complete integrability. Canonical
coordinates are derived, with the map now being a canonical transformation with
a sequence of commuting invariant functions. Compatibility of a pair of these
functions gives rise to Liouville's equation and the map plays the role of a
B\"acklund transformation.Comment: 17 pages, 7 figures. Corrected typos and updated reference detail
A Nonliearly Dispersive Fifth Order Integrable Equation and its Hierarchy
In this paper, we study the properties of a nonlinearly dispersive integrable
system of fifth order and its associated hierarchy. We describe a Lax
representation for such a system which leads to two infinite series of
conserved charges and two hierarchies of equations that share the same
conserved charges. We construct two compatible Hamiltonian structures as well
as their Casimir functionals. One of the structures has a single Casimir
functional while the other has two. This allows us to extend the flows into
negative order and clarifies the meaning of two different hierarchies of
positive flows. We study the behavior of these systems under a hodograph
transformation and show that they are related to the Kaup-Kupershmidt and the
Sawada-Kotera equations under appropriate Miura transformations. We also
discuss briefly some properties associated with the generalization of second,
third and fourth order Lax operators.Comment: 11 pages, LaTex, version to be published in Journal of Nonlinear
Mathematical Physics, has expanded discussio
The generalized Kupershmidt deformation for constructing new integrable systems from integrable bi-Hamiltonian systems
Based on the Kupershmidt deformation for any integrable bi-Hamiltonian
systems presented in [4], we propose the generalized Kupershmidt deformation to
construct new systems from integrable bi-Hamiltonian systems, which provides a
nonholonomic perturbation of the bi-Hamiltonian systems. The generalized
Kupershmidt deformation is conjectured to preserve integrability. The
conjecture is verified in a few representative cases: KdV equation, Boussinesq
equation, Jaulent-Miodek equation and Camassa-Holm equation. For these specific
cases, we present a general procedure to convert the generalized Kupershmidt
deformation into the integrable Rosochatius deformation of soliton equation
with self-consistent sources, then to transform it into a -type
bi-Hamiltonian system. By using this generalized Kupershmidt deformation some
new integrable systems are derived. In fact, this generalized Kupershmidt
deformation also provides a new method to construct the integrable Rosochatius
deformation of soliton equation with self-consistent sources.Comment: 21 pages, to appear in Journal of Mathematical Physic
Integrated Lax Formalism for PCM
By solving the first-order algebraic field equations which arise in the dual
formulation of the D=2 principal chiral model (PCM) we construct an integrated
Lax formalism built explicitly on the dual fields of the model rather than the
currents. The Lagrangian of the dual scalar field theory is also constructed.
Furthermore we present the first-order PDE system for an exponential
parametrization of the solutions and discuss the Frobenious integrability of
this system.Comment: 24 page
Painleve property and the first integrals of nonlinear differential equations
Link between the Painleve property and the first integrals of nonlinear
ordinary differential equations in polynomial form is discussed. The form of
the first integrals of the nonlinear differential equations is shown to
determine by the values of the Fuchs indices. Taking this idea into
consideration we present the algorithm to look for the first integrals of the
nonlinear differential equations in the polynomial form. The first integrals of
five nonlinear ordinary differential equations are found. The general solution
of one of the fourth ordinary differential equations is given.Comment: 22 page
Eisenhart lift of 2-dimensional mechanics
The Eisenhart lift is a variant of geometrization of classical mechanics with d degrees of freedom in which the equations of motion are embedded into the geodesic equations of a Brinkmann-type metric defined on (d+2)
(d+2)
-dimensional spacetime of Lorentzian signature. In this work, the Eisenhart lift of 2-dimensional mechanics on curved background is studied. The corresponding 4-dimensional metric is governed by two scalar functions which are just the conformal factor and the potential of the original dynamical system. We derive a conformal symmetry and a corresponding quadratic integral, associated with the Eisenhart lift. The energy–momentum tensor is constructed which, along with the metric, provides a solution to the Einstein equations. Uplifts of 2-dimensional superintegrable models are discussed with a particular emphasis on the issue of hidden symmetries. It is shown that for the 2-dimensional Darboux–Koenigs metrics, only type I can result in Eisenhart lifts which satisfy the weak energy condition. However, some physically viable metrics with hidden symmetries are presented
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
Solitons from Dressing in an Algebraic Approach to the Constrained KP Hierarchy
The algebraic matrix hierarchy approach based on affine Lie algebras
leads to a variety of 1+1 soliton equations. By varying the rank of the
underlying algebra as well as its gradation in the affine setting, one
encompasses the set of the soliton equations of the constrained KP hierarchy.
The soliton solutions are then obtained as elements of the orbits of the
dressing transformations constructed in terms of representations of the vertex
operators of the affine algebras realized in the unconventional
gradations. Such soliton solutions exhibit non-trivial dependence on the KdV
(odd) time flows and KP (odd and even) time flows which distinguishes them from
the conventional structure of the Darboux-B\"{a}cklund Wronskian solutions of
the constrained KP hierarchy.Comment: LaTeX, 13pg
- …