1,031 research outputs found
On the calculation of finite-gap solutions of the KdV equation
A simple and general approach for calculating the elliptic finite-gap
solutions of the Korteweg-de Vries (KdV) equation is proposed. Our approach is
based on the use of the finite-gap equations and the general representation of
these solutions in the form of rational functions of the elliptic Weierstrass
function. The calculation of initial elliptic finite-gap solutions is reduced
to the solution of the finite-band equations with respect to the parameters of
the representation. The time evolution of these solutions is described via the
dynamic equations of their poles, integrated with the help of the finite-gap
equations. The proposed approach is applied by calculating the elliptic 1-, 2-
and 3-gap solutions of the KdV equations
A precise definition of reduction of partial differential equations
We give a comprehensive analysis of interrelations between the basic concepts
of the modern theory of symmetry (classical and non-classical) reductions of
partial differential equations. Using the introduced definition of reduction of
differential equations we establish equivalence of the non-classical
(conditional symmetry) and direct (Ansatz) approaches to reduction of partial
differential equations. As an illustration we give an example of non-classical
reduction of the nonlinear wave equation in (1+3) dimensions. The conditional
symmetry approach when applied to the equation in question yields a number of
non-Lie reductions which are far-reaching generalization of the well-known
symmetry reductions of the nonlinear wave equations.Comment: LaTeX, 21 page
Noncompact SL(2,R) spin chain
We consider the integrable spin chain model - the noncompact SL(2,R) spin
magnet. The spin operators are realized as the generators of the unitary
principal series representation of the SL(2,R) group. In an explicit form, we
construct R-matrix, the Baxter Q-operator and the transition kernel to the
representation of the Separated Variables (SoV). The expressions for the energy
and quasimomentum of the eigenstates in terms of the Baxter Q-operator are
derived. The analytic properties of the eigenvalues of the Baxter operator as a
function of the spectral parameter are established. Applying the diagrammatic
approach, we calculate Sklyanin's integration measure in the separated
variables and obtain the solution to the spectral problem for the model in
terms of the eigenvalues of the Q-operator. We show that the transition kernel
to the SoV representation is factorized into a product of certain operators
each depending on a single separated variable.Comment: 29 pages, 12 figure
On Darboux-Treibich-Verdier potentials
It is shown that the four-parameter family of elliptic functions
introduced
by Darboux and rediscovered a hundred years later by Treibich and Verdier, is
the most general meromorphic family containing infinitely many finite-gap
potentials.Comment: 8 page
Variational Approximations in a Path-Integral Description of Potential Scattering
Using a recent path integral representation for the T-matrix in
nonrelativistic potential scattering we investigate new variational
approximations in this framework. By means of the Feynman-Jensen variational
principle and the most general ansatz quadratic in the velocity variables --
over which one has to integrate functionally -- we obtain variational equations
which contain classical elements (trajectories) as well as quantum-mechanical
ones (wave spreading).We analyse these equations and solve them numerically by
iteration, a procedure best suited at high energy. The first correction to the
variational result arising from a cumulant expansion is also evaluated.
Comparison is made with exact partial-wave results for scattering from a
Gaussian potential and better agreement is found at large scattering angles
where the standard eikonal-type approximations fail.Comment: 35 pages, 3 figures, 6 tables, Latex with amsmath, amssymb; v2: 28
pages, EPJ style, misprints corrected, note added about correct treatment of
complex Gaussian integrals with the theory of "pencils", matches published
versio
Wilson function transforms related to Racah coefficients
The irreducible -representations of the Lie algebra consist of
discrete series representations, principal unitary series and complementary
series. We calculate Racah coefficients for tensor product representations that
consist of at least two discrete series representations. We use the explicit
expressions for the Clebsch-Gordan coefficients as hypergeometric functions to
find explicit expressions for the Racah coefficients. The Racah coefficients
are Wilson polynomials and Wilson functions. This leads to natural
interpretations of the Wilson function transforms. As an application several
sum and integral identities are obtained involving Wilson polynomials and
Wilson functions. We also compute Racah coefficients for U_q(\su(1,1)), which
turn out to be Askey-Wilson functions and Askey-Wilson polynomials.Comment: 48 page
Pion, kaon, proton and anti-proton transverse momentum distributions from p+p and d+Au collisions at GeV
Identified mid-rapidity particle spectra of , , and
from 200 GeV p+p and d+Au collisions are reported. A
time-of-flight detector based on multi-gap resistive plate chamber technology
is used for particle identification. The particle-species dependence of the
Cronin effect is observed to be significantly smaller than that at lower
energies. The ratio of the nuclear modification factor () between
protons and charged hadrons () in the transverse momentum
range GeV/c is measured to be
(stat)(syst) in minimum-bias collisions and shows little
centrality dependence. The yield ratio of in minimum-bias d+Au
collisions is found to be a factor of 2 lower than that in Au+Au collisions,
indicating that the Cronin effect alone is not enough to account for the
relative baryon enhancement observed in heavy ion collisions at RHIC.Comment: 6 pages, 4 figures, 1 table. We extended the pion spectra from
transverse momentum 1.8 GeV/c to 3. GeV/
Kaon Production and Kaon to Pion Ratio in Au+Au Collisions at \snn=130 GeV
Mid-rapidity transverse mass spectra and multiplicity densities of charged
and neutral kaons are reported for Au+Au collisions at \snn=130 GeV at RHIC.
The spectra are exponential in transverse mass, with an inverse slope of about
280 MeV in central collisions. The multiplicity densities for these particles
scale with the negative hadron pseudo-rapidity density. The charged kaon to
pion ratios are and
for the most central collisions. The ratio is lower than the same
ratio observed at the SPS while the is higher than the SPS result.
Both ratios are enhanced by about 50% relative to p+p and +p
collision data at similar energies.Comment: 6 pages, 3 figures, 1 tabl
Azimuthal anisotropy and correlations in p+p, d+Au and Au+Au collisions at 200 GeV
We present the first measurement of directed flow () at RHIC. is
found to be consistent with zero at pseudorapidities from -1.2 to 1.2,
then rises to the level of a couple of percent over the range . The latter observation is similar to data from NA49 if the SPS rapidities
are shifted by the difference in beam rapidity between RHIC and SPS.
Back-to-back jets emitted out-of-plane are found to be suppressed more if
compared to those emitted in-plane, which is consistent with {\it jet
quenching}. Using the scalar product method, we systematically compared
azimuthal correlations from p+p, d+Au and Au+Au collisions. Flow and non-flow
from these three different collision systems are discussed.Comment: Quark Matter 2004 proceeding, 4 pages, 3 figure
Azimuthal anisotropy: the higher harmonics
We report the first observations of the fourth harmonic (v_4) in the
azimuthal distribution of particles at RHIC. The measurement was done taking
advantage of the large elliptic flow generated at RHIC. The integrated v_4 is
about a factor of 10 smaller than v_2. For the sixth (v_6) and eighth (v_8)
harmonics upper limits on the magnitudes are reported.Comment: 4 pages, 6 figures, contribution to the Quark Matter 2004 proceeding
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