695 research outputs found
Continuous and discontinuous piecewise linear solutions of the linearly forced inviscid Burgers equation
We study a class of piecewise linear solutions to the inviscid Burgers
equation driven by a linear forcing term. Inspired by the analogy with peakons,
we think of these solutions as being made up of solitons situated at the
breakpoints. We derive and solve ODEs governing the soliton dynamics, first for
continuous solutions, and then for more general shock wave solutions with
discontinuities. We show that triple collisions of solitons cannot take place
for continuous solutions, but give an example of a triple collision in the
presence of a shock.Comment: To appear in Journal of Nonlinear Mathematical Physics (proceedings
of NEEDS 2007). 16 pages, 3 figures, LaTeX + AMS packages + pstrick
Tunneling Theory for Tunable Open Quantum Systems of Ultracold Atoms in One-Dimensional Traps
The creation of tunable open quantum systems is becoming feasible in current
experiments with ultracold atoms in low-dimensional traps. In particular, the
high degree of experimental control over these systems allows detailed studies
of tunneling dynamics, e.g., as a function of the trapping geometry and the
interparticle interaction strength. In order to address this exciting
opportunity we present a theoretical framework for two-body tunneling based on
the rigged Hilbert space formulation. In this approach, bound, resonant and
scattering states are included on an equal footing, and we argue that the
coupling of all these components is vital for a correct description of the
relevant threshold phenomena. In particular, we study the tunneling mechanism
for two-body systems in one-dimensional traps and different interaction
regimes. We find a strong dominance of sequential tunneling of single particles
for repulsive and weakly attractive systems, while there is a signature of
correlated pair tunneling in the calculated many-particle flux for strongly
attractive interparticle interaction.Comment: To be published in Phys. Rev. A (Rapid Communication
The inverse spectral problem for the discrete cubic string
Given a measure on the real line or a finite interval, the "cubic string"
is the third order ODE where is a spectral parameter. If
equipped with Dirichlet-like boundary conditions this is a nonselfadjoint
boundary value problem which has recently been shown to have a connection to
the Degasperis-Procesi nonlinear water wave equation. In this paper we study
the spectral and inverse spectral problem for the case of Neumann-like boundary
conditions which appear in a high-frequency limit of the Degasperis--Procesi
equation. We solve the spectral and inverse spectral problem for the case of
being a finite positive discrete measure. In particular, explicit
determinantal formulas for the measure are given. These formulas generalize
Stieltjes' formulas used by Krein in his study of the corresponding second
order ODE .Comment: 24 pages. LaTeX + iopart, xypic, amsthm. To appear in Inverse
Problems (http://www.iop.org/EJ/journal/IP
The Cauchy two-matrix model
We introduce a new class of two(multi)-matrix models of positive Hermitean
matrices coupled in a chain; the coupling is related to the Cauchy kernel and
differs from the exponential coupling more commonly used in similar models. The
correlation functions are expressed entirely in terms of certain biorthogonal
polynomials and solutions of appropriate Riemann-Hilbert problems, thus paving
the way to a steepest descent analysis and universality results. The
interpretation of the formal expansion of the partition function in terms of
multicolored ribbon-graphs is provided and a connection to the O(1) model. A
steepest descent analysis of the partition function reveals that the model is
related to a trigonal curve (three-sheeted covering of the plane) much in the
same way as the Hermitean matrix model is related to a hyperelliptic curve.Comment: 34 pages, 2 figures. V2: changes only to metadat
Cauchy Biorthogonal Polynomials
The paper investigates the properties of certain biorthogonal polynomials
appearing in a specific simultaneous Hermite-Pade' approximation scheme.
Associated to any totally positive kernel and a pair of positive measures on
the positive axis we define biorthogonal polynomials and prove that their
zeroes are simple and positive. We then specialize the kernel to the Cauchy
kernel 1/{x+y} and show that the ensuing biorthogonal polynomials solve a
four-term recurrence relation, have relevant Christoffel-Darboux generalized
formulae, and their zeroes are interlaced. In addition, these polynomial solve
a combination of Hermite-Pade' approximation problems to a Nikishin system of
order 2. The motivation arises from two distant areas; on one side, in the
study of the inverse spectral problem for the peakon solution of the
Degasperis-Procesi equation; on the other side, from a random matrix model
involving two positive definite random Hermitian matrices. Finally, we show how
to characterize these polynomials in term of a Riemann-Hilbert problem.Comment: 38 pages, partially replaces arXiv:0711.408
Testing the Hubble Law with the IRAS 1.2 Jy Redshift Survey
We test and reject the claim of Segal et al. (1993) that the correlation of
redshifts and flux densities in a complete sample of IRAS galaxies favors a
quadratic redshift-distance relation over the linear Hubble law. This is done,
in effect, by treating the entire galaxy luminosity function as derived from
the 60 micron 1.2 Jy IRAS redshift survey of Fisher et al. (1995) as a distance
indicator; equivalently, we compare the flux density distribution of galaxies
as a function of redshift with predictions under different redshift-distance
cosmologies, under the assumption of a universal luminosity function. This
method does not assume a uniform distribution of galaxies in space. We find
that this test has rather weak discriminatory power, as argued by Petrosian
(1993), and the differences between models are not as stark as one might expect
a priori. Even so, we find that the Hubble law is indeed more strongly
supported by the analysis than is the quadratic redshift-distance relation. We
identify a bias in the the Segal et al. determination of the luminosity
function, which could lead one to mistakenly favor the quadratic
redshift-distance law. We also present several complementary analyses of the
density field of the sample; the galaxy density field is found to be close to
homogeneous on large scales if the Hubble law is assumed, while this is not the
case with the quadratic redshift-distance relation.Comment: 27 pages Latex (w/figures), ApJ, in press. Uses AAS macros,
postscript also available at
http://www.astro.princeton.edu/~library/preprints/pop682.ps.g
Inverse problems associated with integrable equations of Camassa-Holm type; explicit formulas on the real axis, I
The inverse problem which arises in the Camassa--Holm equation is revisited
for the class of discrete densities. The method of solution relies on the use
of orthogonal polynomials. The explicit formulas are obtained directly from the
analysis on the real axis without any additional transformation to a "string"
type boundary value problem known from prior works
The Degasperis-Procesi equation with self-consistent sources
The Degasperis-Procesi equation with self-consistent sources(DPESCS) is
derived. The Lax representation and the conservation laws for DPESCS are
constructed. The peakon solution of DPESCS is obtained.Comment: 15 page
Projective dynamics and first integrals
We present the theory of tensors with Young tableau symmetry as an efficient
computational tool in dealing with the polynomial first integrals of a natural
system in classical mechanics. We relate a special kind of such first
integrals, already studied by Lundmark, to Beltrami's theorem about
projectively flat Riemannian manifolds. We set the ground for a new and simple
theory of the integrable systems having only quadratic first integrals. This
theory begins with two centered quadrics related by central projection, each
quadric being a model of a space of constant curvature. Finally, we present an
extension of these models to the case of degenerate quadratic forms.Comment: 39 pages, 2 figure
A class of Poisson-Nijenhuis structures on a tangent bundle
Equipping the tangent bundle TQ of a manifold with a symplectic form coming
from a regular Lagrangian L, we explore how to obtain a Poisson-Nijenhuis
structure from a given type (1,1) tensor field J on Q. It is argued that the
complete lift of J is not the natural candidate for a Nijenhuis tensor on TQ,
but plays a crucial role in the construction of a different tensor R, which
appears to be the pullback under the Legendre transform of the lift of J to
co-tangent manifold of Q. We show how this tangent bundle view brings new
insights and is capable also of producing all important results which are known
from previous studies on the cotangent bundle, in the case that Q is equipped
with a Riemannian metric. The present approach further paves the way for future
generalizations.Comment: 22 page
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