727 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
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
Multi loop soliton solutions and their interactions in the Degasperis-Procesi equation
In this article, we construct loop soliton solutions and mixed soliton - loop
soliton solution for the Degasperis-Procesi equation. To explore these
solutions we adopt the procedure given by Matsuno. By appropriately modifying
the -function given in the above paper we derive these solutions. We
present the explicit form of one and two loop soliton solutions and mixed
soliton - loop soliton solutions and investigate the interaction between (i)
two loop soliton solutions in different parametric regimes and (ii) a loop
soliton with a conventional soliton in detail.Comment: Published in Physica Scripta (2012
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
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
Does Surgical Resection of Horizontal Extraocular Muscles Disrupt Ocular Proprioceptors? [Corrigendum]
Paduca A, Lundmark PO, Bruenech JR. Clin Ophthalmol. 2023;17:1395-1405.
Page 1396, Materials and Methods, first paragraph, line 4 the text “recession/resection surgery” should read “recession/resection surgery for the first time”.
Page 1396, Materials and Methods, first paragraph, line 5, the text “intermittent” should read “intermittent with alphabetical pattern”.
The authors apologize for this error
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
Peakons arising as particle paths beneath small-amplitude water waves
We present a new kind of particle path in constant vorticity water of finite
depth, within the framework of small-amplitude waves
On the tau-functions of the Degasperis-Procesi equation
The DP equation is investigated from the point of view of
determinant-pfaffian identities. The reciprocal link between the
Degasperis-Procesi (DP) equation and the pseudo 3-reduction of the
two-dimensional Toda system is used to construct the N-soliton solution of the
DP equation. The N-soliton solution of the DP equation is presented in the form
of pfaffian through a hodograph (reciprocal) transformation. The bilinear
equations, the identities between determinants and pfaffians, and the
-functions of the DP equation are obtained from the pseudo 3-reduction of
the two-dimensional Toda system.Comment: 27 pages, 4 figures, Journal of Physics A: Mathematical and
Theoretical, to be publishe
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