6,541 research outputs found
Singular structure of Toda lattices and cohomology of certain compact Lie groups
We study the singularities (blow-ups) of the Toda lattice associated with a
real split semisimple Lie algebra . It turns out that the total
number of blow-up points along trajectories of the Toda lattice is given by the
number of points of a Chevalley group related to the maximal
compact subgroup of the group with over the finite field . Here is the Langlands dual of . The blow-ups of the Toda lattice
are given by the zero set of the -functions. For example, the blow-ups of
the Toda lattice of A-type are determined by the zeros of the Schur polynomials
associated with rectangular Young diagrams. Those Schur polynomials are the
-functions for the nilpotent Toda lattices. Then we conjecture that the
number of blow-ups is also given by the number of real roots of those Schur
polynomials for a specific variable. We also discuss the case of periodic Toda
lattice in connection with the real cohomology of the flag manifold associated
to an affine Kac-Moody algebra.Comment: 23 pages, 12 figures, To appear in the proceedings "Topics in
Integrable Systems, Special Functions, Orthogonal Polynomials and Random
Matrices: Special Volume, Journal of Computational and Applied Mathematics
Confluence of hypergeometric functions and integrable hydrodynamic type systems
It is known that a large class of integrable hydrodynamic type systems can be
constructed through the Lauricella function, a generalization of the classical
Gauss hypergeometric function. In this paper, we construct novel class of
integrable hydrodynamic type systems which govern the dynamics of critical
points of confluent Lauricella type functions defined on finite dimensional
Grassmannian Gr(2,n), the set of 2xn matrices of rank two. Those confluent
functions satisfy certain degenerate Euler-Poisson-Darboux equations. It is
also shown that in general, hydrodynamic type system associated to the
confluent Lauricella function is given by an integrable and non-diagonalizable
quasi-linear system of a Jordan matrix form. The cases of Grassmannian Gr(2,5)
for two component systems and Gr(2,6) for three component systems are
considered in details.Comment: 22 pages, PMNP 2015, added some comments and reference
Perturbative analysis of wave interactions in nonlinear systems
This work proposes a new way for handling obstacles to asymptotic
integrability in perturbed nonlinear PDEs within the method of Normal Forms -
NF - for the case of multi-wave solutions. Instead of including the whole
obstacle in the NF, only its resonant part is included, and the remainder is
assigned to the homological equation. This leaves the NF intergable and its
solutons retain the character of the solutions of the unperturbed equation. We
exploit the freedom in the expansion to construct canonical obstacles which are
confined to te interaction region of the waves. Fo soliton solutions, e.g., in
the KdV equation, the interaction region is a finite domain around the origin;
the canonical obstacles then do not generate secular terms in the homological
equation. When the interaction region is infifnite, or semi-infinite, e.g., in
wave-front solutions of the Burgers equation, the obstacles may contain
resonant terms. The obstacles generate waves of a new type, which cannot be
written as functionals of the solutions of the NF. When an obstacle contributes
a resonant term to the NF, this leads to a non-standard update of th wave
velocity.Comment: 13 pages, including 6 figure
HBT Interferometry for Sonoluminescence Bubble
The two-photon correlation of the light pulse emitted from a sonoluminescence
bubble is discussed. It is shown that several important information about the
mechanism of light emission, such as the time-scale and the shape of the
emission region could be obtained from the HBT interferometry. We also argue
that such a measurement may serve to reject one of the two currently suggested
emission mechanisms, i.e., thermal process versus dynamical Casimir effect.Comment: 13 pages, RevTeX, 2 eps figures include
Topics on Hydrodynamic Model of Nucleus-Nucleus Collisions
A survey is given on the applications of hydrodynamic model of
nucleus-nucleus collisons, focusing especially on i) the resolution of
hydrodynamic equations for arbitrary configurations, by using the
smoothed-particle hydrodynamic approach; ii) effects of the event-by-event
fluctuation of the initial conditions on the observables; iii) decoupling
criteria; iv) analytical solutions; and others.Comment: 30 pages, 29 figures; corrected typo
- …