1,783 research outputs found
Deformations of the Monge/Riemann hierarchy and approximately integrable systems
Dispersive deformations of the Monge equation u_u=uu_x are studied using
ideas originating from topological quantum field theory and the deformation
quantization programme. It is shown that, to a high-order, the symmetries of
the Monge equation may also be appropriately deformed, and that, if they exist
at all orders, they are uniquely determined by the original deformation. This
leads to either a new class of integrable systems or to a rigorous notion of an
approximate integrable system. Quasi-Miura transformations are also constructed
for such deformed equations.Comment: 9 pages LaTe
Symmetries and Solutions of Getzler's Equation for Coxeter and Extended Affine Weyl Frobenius Manifolds
The G-function associated to the semi-simple Frobenius manifold C^n/W (where
W is a Coxeter group or an extended affine Weyl group) is studied. The general
form of the G function is given in terms of a logarithmic singularity over
caustics in the manifold. The main result in this paper is a universal formula
for the G-function corresponding to the Frobenius manifold C^n/W^(k)(A_{n-1})
where W^(k)(A_{n-1}) is a certain extended affine Weyl group (or, equivalently,
corresponding to the Hurwitz space M_{0;k-1,n-k-1}), together with the general
form of the G-function in terms of data on caustics. Symmetries of the G
function are also studied.Comment: 9 pages, LaTe
Simple Elliptic Singularities: a note on their G-function
The link between Frobenius manifolds and singularity theory is well known,
with the simplest examples coming from the simple hypersurface singularities.
Associated with any such manifold is a function known as the -function. This
plays a role in the construction of higher-genus terms in various theories. For
the simple singularities the G-function is known explicitly: G=0. The next
class of singularities, the unimodal hypersurface or elliptic hypersurface
singularities consists of three examples,
\widetilde{E}_6,\widetilde{E}_7,\widetilde{E}_8 (or equivalently P_8,
X_9,J_10). Using a result of Noumi and Yamada on the flat structure on the
space of versal deformations of these singularities the -function is
explicitly constructed for these three examples. The main property is that the
function depends on only one variable, the marginal (dimensionless) deformation
variable. Other examples are given based on the foldings of known Frobenius
manifolds. Properties of the -function under the action of the modular group
is studied, and applications within the theory of integrable systems are
discussed.Comment: 15 page
A Geometry for Multidimensional Integrable Systems
A deformed differential calculus is developed based on an associative
star-product. In two dimensions the Hamiltonian vector fields model the algebra
of pseudo-differential operator, as used in the theory of integrable systems.
Thus one obtains a geometric description of the operators. A dual theory is
also possible, based on a deformation of differential forms. This calculus is
applied to a number of multidimensional integrable systems, such as the KP
hierarchy, thus obtaining a geometrical description of these systems. The limit
in which the deformation disappears corresponds to taking the dispersionless
limit in these hierarchies.Comment: LaTeX, 29 pages. To be published in J.Geom.Phy
Deformations of dispersionless KdV hierarchies
The obstructions to the existence of a hierarchy of hydrodynamic conservation
laws are studied for a multicomponent dispersionless KdV system. It is shown
that if an underlying algebra is Jordan, then the lowest obstruction vanishes
and that all higher obstructions automatically vanish. Deformations of these
multicomponent dispersionless KdV-type equations are also studied. No new
obstructions appear, and hence the existence of a fully deformed hierarchy
depends on the existence of a single purely hydrodynamic conservation law.Comment: 12 papge
Degenerate Frobenius manifolds and the bi-Hamiltonian structure of rational Lax equations
The bi-Hamiltonian structure of certain multi-component integrable systems,
generalizations of the dispersionless Toda hierarchy, is studies for systems
derived from a rational Lax function. One consequence of having a rational
rather than a polynomial Lax function is that the corresponding bi-Hamiltonian
structures are degenerate, i.e. the metric which defines the Hamiltonian
structure has vanishing determinant. Frobenius manifolds provide a natural
setting in which to study the bi-Hamiltonian structure of certain classes of
hydrodynamic systems. Some ideas on how this structure may be extanded to
include degenerate bi-Hamiltonian structures, such as those given in the first
part of the paper, are given.Comment: 28 pages, LaTe
On the isomonodromic tau-function for the Hurwitz spaces of branched coverings of genus zero and one
The isomonodromic tau-function for the Hurwitz spaces of branched coverings
of genus zero and one are constructed explicitly. Such spaces may be equipped
with the structure of a Frobenius manifold and this introduces a flat
coordinate system on the manifold. The isomonodromic tau-function, and in
particular the associated -function, are rewritten in these coordinates and
an interpretation in terms of the caustics (where the multiplication is not
semisimple) is given.Comment: 18 page
The dispersive self-dual Einstein equations and the Toda lattice
The Boyer-Finley equation, or -Toda equation is both a reduction
of the self-dual Einstein equations and the dispersionlesslimit of the
-Toda lattice equation. This suggests that there should be a dispersive
version of the self-dual Einstein equation which both contains the Toda lattice
equation and whose dispersionless limit is the familiar self-dual Einstein
equation. Such a system is studied in this paper. The results are achieved by
using a deformation, based on an associative -product, of the algebra
used in the study of the undeformed, or dispersionless,
equations.Comment: 11 pages, LaTeX. To appear: J. Phys.
A note on the relationship between rational and trigonometric solutions of the WDVV equations
Legendre transformations provide a natural symmetry on the space of solutions to the WDVV equations, and more specifically, between different Frobenius manifolds. In this paper a twisted Legendre transformation is constructed between solutions which define the corresponding dual Frobenius manifolds. As an application it is shown that certain trigonometric and rational solutions of the WDVV equations are related by such a twisted Legendre transform
The algebraic and Hamiltonian structure of the dispersionless Benney and Toda hierarchies
The algebraic and Hamiltonian structures of the multicomponent dispersionless
Benney and Toda hierarchies are studied. This is achieved by using a modified
set of variables for which there is a symmetry between the basic fields. This
symmetry enables formulae normally given implicitly in terms of residues, such
as conserved charges and fluxes, to be calculated explicitly. As a corollary of
these results the equivalence of the Benney and Toda hierarchies is
established. It is further shown that such quantities may be expressed in terms
of generalized hypergeometric functions, the simplest example involving
Legendre polynomials. These results are then extended to systems derived from a
rational Lax function and a logarithmic function. Various reductions are also
studied.Comment: 29 pages, LaTe
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