123 research outputs found

### On Darboux-Treibich-Verdier potentials

It is shown that the four-parameter family of elliptic functions
$u_D(z)=m_0(m_0+1)\wp(z)+\sum_{i=1}^3 m_i(m_i+1)\wp(z-\omega_i)$ introduced
by Darboux and rediscovered a hundred years later by Treibich and Verdier, is
the most general meromorphic family containing infinitely many finite-gap
potentials.Comment: 8 page

### On the Equivalence of Different Lax Pairs for the Kac-van Moerbeke Hierarchy

We give a simple algebraic proof that the two different Lax pairs for the
Kac-van Moerbeke hierarchy, constructed from Jacobi respectively
super-symmetric Dirac-type difference operators, give rise to the same
hierarchy of evolution equations. As a byproduct we obtain some new recursions
for computing these equations.Comment: 8 page

### On the calculation of finite-gap solutions of the KdV equation

A simple and general approach for calculating the elliptic finite-gap
solutions of the Korteweg-de Vries (KdV) equation is proposed. Our approach is
based on the use of the finite-gap equations and the general representation of
these solutions in the form of rational functions of the elliptic Weierstrass
function. The calculation of initial elliptic finite-gap solutions is reduced
to the solution of the finite-band equations with respect to the parameters of
the representation. The time evolution of these solutions is described via the
dynamic equations of their poles, integrated with the help of the finite-gap
equations. The proposed approach is applied by calculating the elliptic 1-, 2-
and 3-gap solutions of the KdV equations

### Hamiltonian systems of hydrodynamic type in 2 + 1 dimensions

We investigate multi-dimensional Hamiltonian systems associated with constant
Poisson brackets of hydrodynamic type. A complete list of two- and
three-component integrable Hamiltonians is obtained. All our examples possess
dispersionless Lax pairs and an infinity of hydrodynamic reductions.Comment: 34 page

### Integrable Systems and Metrics of Constant Curvature

In this article we present a Lagrangian representation for evolutionary
systems with a Hamiltonian structure determined by a differential-geometric
Poisson bracket of the first order associated with metrics of constant
curvature. Kaup-Boussinesq system has three local Hamiltonian structures and
one nonlocal Hamiltonian structure associated with metric of constant
curvature. Darboux theorem (reducing Hamiltonian structures to canonical form
''d/dx'' by differential substitutions and reciprocal transformations) for
these Hamiltonian structures is proved

### Axial anomaly with the overlap-Dirac operator in arbitrary dimensions

We evaluate for arbitrary even dimensions the classical continuum limit of
the lattice axial anomaly defined by the overlap-Dirac operator. Our
calculational scheme is simple and systematic. In particular, a powerful
topological argument is utilized to determine the value of a lattice integral
involved in the calculation. When the Dirac operator is free of species
doubling, the classical continuum limit of the axial anomaly in various
dimensions is combined into a form of the Chern character, as expected.Comment: 9 pages, uses JHEP.cls and amsfonts.sty, the final version to appear
in JHE

### Dispersionless Toda hierarchy and two-dimensional string theory

The dispersionless Toda hierarchy turns out to lie in the heart of a recently
proposed Landau-Ginzburg formulation of two-dimensional string theory at
self-dual compactification radius. The dynamics of massless tachyons with
discrete momenta is shown to be encoded into the structure of a special
solution of this integrable hierarchy. This solution is obtained by solving a
Riemann-Hilbert problem. Equivalence to the tachyon dynamics is proven by
deriving recursion relations of tachyon correlation functions in the machinery
of the dispersionless Toda hierarchy. Fundamental ingredients of the
Landau-Ginzburg formulation, such as Landau-Ginzburg potentials and tachyon
Landau-Ginzburg fields, are translated into the language of the Lax formalism.
Furthermore, a wedge algebra is pointed out to exist behind the Riemann-Hilbert
problem, and speculations on its possible role as generators of ``extra''
states and fields are presented.Comment: LaTeX 21 pages, KUCP-0067 (typos are corrected and a brief note is
added

### The Whitham Deformation of the Dijkgraaf-Vafa Theory

We discuss the Whitham deformation of the effective superpotential in the
Dijkgraaf-Vafa (DV) theory. It amounts to discussing the Whitham deformation of
an underlying (hyper)elliptic curve. Taking the elliptic case for simplicity we
derive the Whitham equation for the period, which governs flowings of branch
points on the Riemann surface. By studying the hodograph solution to the
Whitham equation it is shown that the effective superpotential in the DV theory
is realized by many different meromorphic differentials. Depending on which
meromorphic differential to take, the effective superpotential undergoes
different deformations. This aspect of the DV theory is discussed in detail by
taking the N=1^* theory. We give a physical interpretation of the deformation
parameters.Comment: 35pages, 1 figure; v2: one section added to give a physical
interpretation of the deformation parameters, one reference added, minor
corrections; v4: minor correction

### Systems of Hess-Appel'rot type

We construct higher-dimensional generalizations of the classical
Hess-Appel'rot rigid body system. We give a Lax pair with a spectral parameter
leading to an algebro-geometric integration of this new class of systems, which
is closely related to the integration of the Lagrange bitop performed by us
recently and uses Mumford relation for theta divisors of double unramified
coverings. Based on the basic properties satisfied by such a class of systems
related to bi-Poisson structure, quasi-homogeneity, and conditions on the
Kowalevski exponents, we suggest an axiomatic approach leading to what we call
the "class of systems of Hess-Appel'rot type".Comment: 40 pages. Comm. Math. Phys. (to appear

### Isospectral Flow and Liouville-Arnold Integration in Loop Algebras

A number of examples of Hamiltonian systems that are integrable by classical
means are cast within the framework of isospectral flows in loop algebras.
These include: the Neumann oscillator, the cubically nonlinear Schr\"odinger
systems and the sine-Gordon equation. Each system has an associated invariant
spectral curve and may be integrated via the Liouville-Arnold technique. The
linearizing map is the Abel map to the associated Jacobi variety, which is
deduced through separation of variables in hyperellipsoidal coordinates. More
generally, a family of moment maps is derived, identifying certain finite
dimensional symplectic manifolds with rational coadjoint orbits of loop
algebras. Integrable Hamiltonians are obtained by restriction of elements of
the ring of spectral invariants to the image of these moment maps. The
isospectral property follows from the Adler-Kostant-Symes theorem, and gives
rise to invariant spectral curves. {\it Spectral Darboux coordinates} are
introduced on rational coadjoint orbits, generalizing the hyperellipsoidal
coordinates to higher rank cases. Applying the Liouville-Arnold integration
technique, the Liouville generating function is expressed in completely
separated form as an abelian integral, implying the Abel map linearization in
the general case.Comment: 42 pages, 2 Figures, 1 Table. Lectures presented at the VIIIth
Scheveningen Conference, held at Wassenaar, the Netherlands, Aug. 16-21, 199

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