78 research outputs found
Hirota derivatives and representation theory
It is shown that the Hirota derivative can be used to construct the plethysm for tensor products of representations of {sl}_2(k)
On the linearization of the generalized Ermakov systems
A linearization procedure is proposed for Ermakov systems with frequency
depending on dynamic variables. The procedure applies to a wide class of
generalized Ermakov systems which are linearizable in a manner similar to that
applicable to usual Ermakov systems. The Kepler--Ermakov systems belong into
this category but others, more generic, systems are also included
Generalised Elliptic Functions
We consider multiply periodic functions, sometimes called Abelian functions,
defined with respect to the period matrices associated with classes of
algebraic curves. We realise them as generalisations of the Weierstras
P-function using two different approaches. These functions arise naturally as
solutions to some of the important equations of mathematical physics and their
differential equations, addition formulae, and applications have all been
recent topics of study.
The first approach discussed sees the functions defined as logarithmic
derivatives of the sigma-function, a modified Riemann theta-function. We can
make use of known properties of the sigma function to derive power series
expansions and in turn the properties mentioned above. This approach has been
extended to a wide range of non hyperelliptic and higher genus curves and an
overview of recent results is given.
The second approach defines the functions algebraically, after first
modifying the curve into its equivariant form. This approach allows the use of
representation theory to derive a range of results at lower computational cost.
We discuss the development of this theory for hyperelliptic curves and how it
may be extended in the future.Comment: 16 page
Intertwining Laplace Transformations of Linear Partial Differential Equations
We propose a generalization of Laplace transformations to the case of linear
partial differential operators (LPDOs) of arbitrary order in R^n. Practically
all previously proposed differential transformations of LPDOs are particular
cases of this transformation (intertwining Laplace transformation, ILT). We
give a complete algorithm of construction of ILT and describe the classes of
operators in R^n suitable for this transformation.
Keywords: Integration of linear partial differential equations, Laplace
transformation, differential transformationComment: LaTeX, 25 pages v2: minor misprints correcte
Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces
Universal bi-Hamiltonian hierarchies of group-invariant (multicomponent)
soliton equations are derived from non-stretching geometric curve flows
\map(t,x) in Riemannian symmetric spaces , including compact
semisimple Lie groups for , . The derivation
of these soliton hierarchies utilizes a moving parallel frame and connection
1-form along the curve flows, related to the Klein geometry of the Lie group
where is the local frame structure group. The soliton
equations arise in explicit form from the induced flow on the frame components
of the principal normal vector N=\covder{x}\mapder{x} along each curve, and
display invariance under the equivalence subgroup in that preserves the
unit tangent vector T=\mapder{x} in the framing at any point on a curve.
Their bi-Hamiltonian integrability structure is shown to be geometrically
encoded in the Cartan structure equations for torsion and curvature of the
parallel frame and its connection 1-form in the tangent space T_\map M of the
curve flow. The hierarchies include group-invariant versions of sine-Gordon
(SG) and modified Korteweg-de Vries (mKdV) soliton equations that are found to
be universally given by curve flows describing non-stretching wave maps and
mKdV analogs of non-stretching Schrodinger maps on . These results provide
a geometric interpretation and explicit bi-Hamiltonian formulation for many
known multicomponent soliton equations. Moreover, all examples of
group-invariant (multicomponent) soliton equations given by the present
geometric framework can be constructed in an explicit fashion based on Cartan's
classification of symmetric spaces.Comment: Published version, with a clarification to Theorem 4.5 and a
correction to the Hamiltonian flow in Proposition 5.1
Identities for hyperelliptic P-functions of genus one, two and three in covariant form
We give a covariant treatment of the quadratic differential identities
satisfied by the P-functions on the Jacobian of smooth hyperelliptic curves of
genera 1, 2 and 3
Generalized Hamiltonian structures for Ermakov systems
We construct Poisson structures for Ermakov systems, using the Ermakov
invariant as the Hamiltonian. Two classes of Poisson structures are obtained,
one of them degenerate, in which case we derive the Casimir functions. In some
situations, the existence of Casimir functions can give rise to superintegrable
Ermakov systems. Finally, we characterize the cases where linearization of the
equations of motion is possible
Anisotropic Bose-Einstein condensates and completely integrable dynamical systems
A Gaussian ansatz for the wave function of two-dimensional harmonically
trapped anisotropic Bose-Einstein condensates is shown to lead, via a
variational procedure, to a coupled system of two second-order, nonlinear
ordinary differential equations. This dynamical system is shown to be in the
general class of Ermakov systems. Complete integrability of the resulting
Ermakov system is proven. Using the exact solution, collapse of the condensate
is analyzed in detail. Time-dependence of the trapping potential is allowed
Symplectically-invariant soliton equations from non-stretching geometric curve flows
A moving frame formulation of geometric non-stretching flows of curves in the
Riemannian symmetric spaces and is
used to derive two bi-Hamiltonian hierarchies of symplectically-invariant
soliton equations. As main results, multi-component versions of the sine-Gordon
(SG) equation and the modified Korteweg-de Vries (mKdV) equation exhibiting
invariance are obtained along with their bi-Hamiltonian
integrability structure consisting of a shared hierarchy of symmetries and
conservation laws generated by a hereditary recursion operator. The
corresponding geometric curve flows in and
are shown to be described by a non-stretching wave map and a
mKdV analog of a non-stretching Schr\"odinger map.Comment: 39 pages; remarks added on algebraic aspects of the moving frame used
in the constructio
Curve Flows in Lagrange-Finsler Geometry, Bi-Hamiltonian Structures and Solitons
Methods in Riemann-Finsler geometry are applied to investigate bi-Hamiltonian
structures and related mKdV hierarchies of soliton equations derived
geometrically from regular Lagrangians and flows of non-stretching curves in
tangent bundles. The total space geometry and nonholonomic flows of curves are
defined by Lagrangian semisprays inducing canonical nonlinear connections
(N-connections), Sasaki type metrics and linear connections. The simplest
examples of such geometries are given by tangent bundles on Riemannian
symmetric spaces provided with an N-connection structure and an
adapted metric, for which we elaborate a complete classification, and by
generalized Lagrange spaces with constant Hessian. In this approach,
bi-Hamiltonian structures are derived for geometric mechanical models and
(pseudo) Riemannian metrics in gravity. The results yield horizontal/ vertical
pairs of vector sine-Gordon equations and vector mKdV equations, with the
corresponding geometric curve flows in the hierarchies described in an explicit
form by nonholonomic wave maps and mKdV analogs of nonholonomic Schrodinger
maps on a tangent bundle.Comment: latex 2e 50 pages, the manuscript is a Lagrange-Finsler
generalization of the solitonic Riemannian formalism from math-ph/0608024, v3
modified following requests of Editor/Referee of J. Geom. Phys., new
references and discussion provided in Conclusio
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