113 research outputs found

### 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 $t$-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

### Solitons from Dressing in an Algebraic Approach to the Constrained KP Hierarchy

The algebraic matrix hierarchy approach based on affine Lie $sl (n)$ algebras
leads to a variety of 1+1 soliton equations. By varying the rank of the
underlying $sl (n)$ 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 $sl (n)$ 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

### Darboux transformation for the modified Veselov-Novikov equation

A Darboux transformation is constructed for the modified Veselov-Novikov
equation.Comment: Latex file,8 pages, 0 figure

### Chaos around a H\'enon-Heiles-inspired exact perturbation of a black hole

A solution of the Einstein's equations that represents the superposition of a
Schwarszchild black hole with both quadrupolar and octopolar terms describing a
halo is exhibited. We show that this solution, in the Newtonian limit, is an
analog to the well known H\'enon-Heiles potential. The integrability of orbits
of test particles moving around a black hole representing the galactic center
is studied and bounded zones of chaotic behavior are found.Comment: 7 pages Revte

### On the Caudrey-Beals-Coifman System and the Gauge Group Action

The generalized Zakharov-Shabat systems with complex-valued Cartan elements
and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their
gauge equivalent are studies. This includes: the properties of fundamental
analytical solutions (FAS) for the gauge-equivalent to CBC systems and the
minimal set of scattering data; the description of the class of nonlinear
evolutionary equations solvable by the inverse scattering method and the
recursion operator, related to such systems; the hierarchies of Hamiltonian
structures.Comment: 12 pages, no figures, contribution to the NEEDS 2007 proceedings
(Submitted to J. Nonlin. Math. Phys.

### A Riemann-Hilbert Problem for an Energy Dependent Schr\"odinger Operator

\We consider an inverse scattering problem for Schr\"odinger operators with
energy dependent potentials. The inverse problem is formulated as a
Riemann-Hilbert problem on a Riemann surface. A vanishing lemma is proved for
two distinct symmetry classes. As an application we prove global existence
theorems for the two distinct systems of partial differential equations
$u_t+(u^2/2+w)_x=0, w_t\pm u_{xxx}+(uw)_x=0$ for suitably restricted,
complementary classes of initial data

### Completeness of the cubic and quartic H\'enon-Heiles Hamiltonians

The quartic H\'enon-Heiles Hamiltonian $H = (P_1^2+P_2^2)/2+(\Omega_1
Q_1^2+\Omega_2 Q_2^2)/2
+C Q_1^4+ B Q_1^2 Q_2^2 + A Q_2^4
+(1/2)(\alpha/Q_1^2+\beta/Q_2^2) - \gamma Q_1$ passes the Painlev\'e test for
only four sets of values of the constants. Only one of these, identical to the
traveling wave reduction of the Manakov system, has been explicitly integrated
(Wojciechowski, 1985), while the three others are not yet integrated in the
generic case $(\alpha,\beta,\gamma)\not=(0,0,0)$. We integrate them by building
a birational transformation to two fourth order first degree equations in the
classification (Cosgrove, 2000) of such polynomial equations which possess the
Painlev\'e property. This transformation involves the stationary reduction of
various partial differential equations (PDEs). The result is the same as for
the three cubic H\'enon-Heiles Hamiltonians, namely, in all four quartic cases,
a general solution which is meromorphic and hyperelliptic with genus two. As a
consequence, no additional autonomous term can be added to either the cubic or
the quartic Hamiltonians without destroying the Painlev\'e integrability
(completeness property).Comment: 10 pages, To appear, Theor.Math.Phys. Gallipoli, 34 June--3 July 200

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