343 research outputs found
Tau-Functions generating the Conservation Laws for Generalized Integrable Hierarchies of KdV and Affine-Toda type
For a class of generalized integrable hierarchies associated with affine
(twisted or untwisted) Kac-Moody algebras, an explicit representation of their
local conserved densities by means of a single scalar tau-function is deduced.
This tau-function acts as a partition function for the conserved densities,
which fits its potential interpretation as the effective action of some quantum
system. The class consists of multi-component generalizations of the
Drinfel'd-Sokolov and the two-dimensional affine Toda lattice hierarchies. The
relationship between the former and the approach of Feigin, Frenkel and
Enriquez to soliton equations of KdV and mKdV type is also discussed. These
results considerably simplify the calculation of the conserved charges carried
by the soliton solutions to the equations of the hierarchy, which is important
to establish their interpretation as particles. By way of illustration, we
calculate the charges carried by a set of constrained KP solitons recently
constructed.Comment: 47 pages, plain TeX with AMS fonts, no figure
Thermal diffusion of supersonic solitons in an anharmonic chain of atoms
We study the non-equilibrium diffusion dynamics of supersonic lattice
solitons in a classical chain of atoms with nearest-neighbor interactions
coupled to a heat bath. As a specific example we choose an interaction with
cubic anharmonicity. The coupling between the system and a thermal bath with a
given temperature is made by adding noise, delta-correlated in time and space,
and damping to the set of discrete equations of motion. Working in the
continuum limit and changing to the sound velocity frame we derive a
Korteweg-de Vries equation with noise and damping. We apply a collective
coordinate approach which yields two stochastic ODEs which are solved
approximately by a perturbation analysis. This finally yields analytical
expressions for the variances of the soliton position and velocity. We perform
Langevin dynamics simulations for the original discrete system which fully
confirm the predictions of our analytical calculations, namely noise-induced
superdiffusive behavior which scales with the temperature and depends strongly
on the initial soliton velocity. A normal diffusion behavior is observed for
very low-energy solitons where the noise-induced phonons also make a
significant contribution to the soliton diffusion.Comment: Submitted to PRE. Changes made: New simulations with a different
method of soliton detection. The results and conclusions are not different
from previous version. New appendixes containing information about the system
energy and soliton profile
Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise
We consider two exit problems for the Korteweg-de Vries equation perturbed by
an additive white in time and colored in space noise of amplitude a. The
initial datum gives rise to a soliton when a=0. It has been proved recently
that the solution remains in a neighborhood of a randomly modulated soliton for
times at least of the order of a^{-2}. We prove exponential upper and lower
bounds for the small noise limit of the probability that the exit time from a
neighborhood of this randomly modulated soliton is less than T, of the same
order in a and T. We obtain that the time scale is exactly the right one. We
also study the similar probability for the exit from a neighborhood of the
deterministic soliton solution. We are able to quantify the gain of eliminating
the secular modes to better describe the persistence of the soliton
Isomonodromic deformations and supersymmetric gauge theories
Seiberg-Witten solutions of four-dimensional supersymmetric gauge theories
possess rich but involved integrable structures. The goal of this paper is to
show that an isomonodromy problem provides a unified framework for
understanding those various features of integrability. The Seiberg-Witten
solution itself can be interpreted as a WKB limit of this isomonodromy problem.
The origin of underlying Whitham dynamics (adiabatic deformation of an
isospectral problem), too, can be similarly explained by a more refined
asymptotic method (multiscale analysis). The case of SU()
supersymmetric Yang-Mills theory without matter is considered in detail for
illustration. The isomonodromy problem in this case is closely related to the
third Painlev\'e equation and its multicomponent analogues. An implicit
relation to t\tbar fusion of topological sigma models is thereby expected.Comment: Several typos are corrected, and a few sentenses are altered. 19 pp +
a list of corrections (page 20), LaTe
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