500 research outputs found
An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion
We study a class of 1+1 quadratically nonlinear water wave equations that
combines the linear dispersion of the Korteweg-deVries (KdV) equation with the
nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation, yet still
preserves integrability via the inverse scattering transform (IST) method.
This IST-integrable class of equations contains both the KdV equation and the
CH equation as limiting cases. It arises as the compatibility condition for a
second order isospectral eigenvalue problem and a first order equation for the
evolution of its eigenfunctions. This integrable equation is shown to be a
shallow water wave equation derived by asymptotic expansion at one order higher
approximation than KdV. We compare its traveling wave solutions to KdV
solitons.Comment: 4 pages, no figure
Extension of Hereditary Symmetry Operators
Two models of candidates for hereditary symmetry operators are proposed and
thus many nonlinear systems of evolution equations possessing infinitely many
commutative symmetries may be generated. Some concrete structures of hereditary
symmetry operators are carefully analyzed on the base of the resulting general
conditions and several corresponding nonlinear systems are explicitly given out
as illustrative examples.Comment: 13 pages, LaTe
Boundary value problems for the stationary axisymmetric Einstein equations: a disk rotating around a black hole
We solve a class of boundary value problems for the stationary axisymmetric
Einstein equations corresponding to a disk of dust rotating uniformly around a
central black hole. The solutions are given explicitly in terms of theta
functions on a family of hyperelliptic Riemann surfaces of genus 4. In the
absence of a disk, they reduce to the Kerr black hole. In the absence of a
black hole, they reduce to the Neugebauer-Meinel disk.Comment: 46 page
First Colonization of a Spectral Outpost in Random Matrix Theory
We describe the distribution of the first finite number of eigenvalues in a
newly-forming band of the spectrum of the random Hermitean matrix model. The
method is rigorously based on the Riemann-Hilbert analysis of the corresponding
orthogonal polynomials. We provide an analysis with an error term of order
N^(-2 h) where 1/h = 2 nu+2 is the exponent of non-regularity of the effective
potential, thus improving even in the usual case the analysis of the pertinent
literature. The behavior of the first finite number of zeroes (eigenvalues)
appearing in the new band is analyzed and connected with the location of the
zeroes of certain Freud polynomials. In general all these newborn zeroes
approach the point of nonregularity at the rate N^(-h) whereas one (a stray
zero) lags behind at a slower rate of approach. The kernels for the correlator
functions in the scaling coordinate near the emerging band are provided
together with the subleading term: in particular the transition between K and
K+1 eigenvalues is analyzed in detail.Comment: 32 pages, 8 figures (typo corrected in Formula 4.13); some reference
added and minor correction
Linearizability of the Perturbed Burgers Equation
We show in this letter that the perturbed Burgers equation is equivalent, through a near-identity transformation and
up to order \epsilon, to a linearizable equation if the condition is satisfied. In the case this
condition is not fulfilled, a normal form for the equation under consideration
is given. Then, to illustrate our results, we make a linearizability analysis
of the equations governing the dynamics of a one-dimensional gas.Comment: 10 pages, RevTeX, no figure
A quantitative framework for exploring exit strategies from the COVID-19 lockdown
Following the highly restrictive measures adopted by many countries for combating the current pandemic, the number of individuals infected by SARS-CoV-2 and the associated number of deaths steadily decreased. This fact, together with the impossibility of maintaining the lockdown indefinitely, raises the crucial question of whether it is possible to design an exit strategy based on quantitative analysis. Guided by rigorous mathematical results, we show that this is indeed possible: we present a robust numerical algorithm which can compute the cumulative number of deaths that will occur as a result of increasing the number of contacts by a given multiple, using as input only the most reliable of all data available during the lockdown, namely the cumulative number of deaths
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