29 research outputs found
Whitham-Toda Hierarchy in the Laplacian Growth Problem
The Laplacian growth problem in the limit of zero surface tension is proved
to be equivalent to finding a particular solution to the dispersionless Toda
lattice hierarchy. The hierarchical times are harmonic moments of the growing
domain. The Laplacian growth equation itself is the quasiclassical version of
the string equation that selects the solution to the hierarchy.Comment: 7 pages, no figures, Talk given at the Workshop NEEDS 99 (Crete,
Greece, June 1999
The master T-operator for the Gaudin model and the KP hierarchy
Following the approach of [arXiv:1112.3310], we construct the master T
-operator for the quantum Gaudin model with twisted boundary conditions and
show that it satisfies the bilinear identity and Hirota equations for the
classical KP hierarchy. We also characterize the class of solutions to the KP
hierarchy that correspond to eigenvalues of the master T-operator and study
dynamics of their zeros as functions of the spectral parameter. This implies a
remarkable connection between the quantum Gaudin model and the classical
Calogero-Moser system of particles.Comment: 56 pages, v2: details added in appendix C, v3: published versio
Dispersionless version of the constrained Toda hierarchy and symmetric radial L\"owner equation
We study the dispersionless version of the recently introduced constrained
Toda hierarchy. Like the Toda lattice itself, it admits three equivalent
formulations: the formulation in terms of Lax equations, the formulation of the
Zakharov-Shabat type and the formulation through the generating equation for
the dispersionless limit of logarithm of the tau-function. We show that the
dispersionless constrained Toda hierarchy describes conformal maps of
reflection-symmetric planar domains to the exterior of the unit disc. We also
find finite-dimensional reductions of the hierarchy and show that they are
characterized by a differential equation of the L\"owner type which we call the
symmetric radial L\"owner equation. It is also shown that solutions to the
symmetric radial L\"owner equation are conformal maps of the exterior of the
unit circle with two symmetric slits to the exterior of the unit circle.Comment: 24 pages, 5 figure
Singular limit of Hele-Shaw flow and dispersive regularization of shock waves
We study a family of solutions to the Saffman-Taylor problem with zero
surface tension at a critical regime. In this regime, the interface develops a
thin singular finger. The flow of an isolated finger is given by the Whitham
equations for the KdV integrable hierarchy. We show that the flow describing
bubble break-off is identical to the Gurevich-Pitaevsky solution for
regularization of shock waves in dispersive media. The method provides a scheme
for the continuation of the flow through singularites.Comment: Some typos corrected, added journal referenc
Unstable fingering patterns of Hele-Shaw flows as a dispersionless limit of the KdV hierarchy
We show that unstable fingering patterns of two dimensional flows of viscous
fluids with open boundary are described by a dispersionless limit of the KdV
hierarchy. In this framework, the fingering instability is linked to a known
instability leading to regularized shock solutions for nonlinear waves, in
dispersive media. The integrable structure of the flow suggests a dispersive
regularization of the finite-time singularities.Comment: Published versio
Macdonald Polynomials from Sklyanin Algebras: A Conceptual Basis for the -Adics-Quantum Group Connection
We establish a previously conjectured connection between -adics and
quantum groups. We find in Sklyanin's two parameter elliptic quantum algebra
and its generalizations, the conceptual basis for the Macdonald polynomials,
which ``interpolate'' between the zonal spherical functions of related real and
\--adic symmetric spaces. The elliptic quantum algebras underlie the
\--Baxter models. We show that in the n \air \infty limit, the Jost
function for the scattering of {\em first} level excitations in the
\--Baxter model coincides with the Harish\--Chandra\--like \--function
constructed from the Macdonald polynomials associated to the root system .
The partition function of the \--Baxter model itself is also expressed in
terms of this Macdonald\--Harish\--Chandra\ \--function, albeit in a less
simple way. We relate the two parameters and of the Macdonald
polynomials to the anisotropy and modular parameters of the Baxter model. In
particular the \--adic ``regimes'' in the Macdonald polynomials correspond
to a discrete sequence of XXZ models. We also discuss the possibility of
``\--deforming'' Euler products.Comment: 25 page
A Hierarchical Array of Integrable Models
Motivated by Harish-Chandra theory, we construct, starting from a simple
CDD\--pole \--matrix, a hierarchy of new \--matrices involving ever
``higher'' (in the sense of Barnes) gamma functions.These new \--matrices
correspond to scattering of excitations in ever more complex integrable
models.From each of these models, new ones are obtained either by
``\--deformation'', or by considering the Selberg-type Euler products of
which they represent the ``infinite place''. A hierarchic array of integrable
models is thus obtained. A remarkable diagonal link in this array is
established.Though many entries in this array correspond to familiar integrable
models, the array also leads to new models. In setting up this array we were
led to new results on the \--gamma function and on the \--deformed
Bloch\--Wigner function.Comment: 18 pages, EFI-92-2
Integrable Structure of Interface Dynamics
We establish the equivalence of a 2D contour dynamics to the dispersionless
limit of the integrable Toda hierarchy constrained by a string equation.
Remarkably, the same hierarchy underlies 2D quantum gravity.Comment: 5 pages, no figures, submitted to Phys. Rev. Lett, typos correcte