352 research outputs found
Monte Carlo simulation of Ising model on directed Barabasi-Albert network
The existence of spontaneous magnetization of Ising spins on directed
Barabasi-Albert networks is investigated with seven neighbors, by using Monte
Carlo simulations. In large systems we see the magnetization for different
temperatures T to decay after a characteristic time tau, which is extrapolated
to diverge at zero temperature.Comment: Error corrected, main conclusion unchanged; for Int. J. Mod. Phys. C
16, issue 4 (2005
Reexamination of scaling in the five-dimensional Ising model
In three dimensions, or more generally, below the upper critical dimension,
scaling laws for critical phenomena seem well understood, for both infinite and
for finite systems. Above the upper critical dimension of four, finite-size
scaling is more difficult.
Chen and Dohm predicted deviation in the universality of the Binder cumulants
for three dimensions and more for the Ising model. This deviation occurs if the
critical point T = Tc is approached along lines of constant A = L*L*(T-Tc)/Tc,
then different exponents a function of system size L are found depending on
whether this constant A is taken as positive, zero, or negative. This effect
was confirmed by Monte Carlo simulations with Glauber and Creutz kinetics.
Because of the importance of this effect and the unclear situation in the
analogous percolation problem, we here reexamine the five-dimensional Glauber
kinetics.Comment: 8 pages including 5 figure
Hierarchy of general invariants for bivariate LPDOs
We study invariants under gauge transformations of linear partial
differential operators on two variables. Using results of BK-factorization, we
construct hierarchy of general invariants for operators of an arbitrary order.
Properties of general invariants are studied and some examples are presented.
We also show that classical Laplace invariants correspond to some particular
cases of general invariants.Comment: to appear in J. "Theor.Math.Phys." in May 200
Singular solutions of a modified two-component Camassa-Holm equation
The Camassa-Holm equation (CH) is a well known integrable equation describing
the velocity dynamics of shallow water waves. This equation exhibits
spontaneous emergence of singular solutions (peakons) from smooth initial
conditions. The CH equation has been recently extended to a two-component
integrable system (CH2), which includes both velocity and density variables in
the dynamics. Although possessing peakon solutions in the velocity, the CH2
equation does not admit singular solutions in the density profile. We modify
the CH2 system to allow dependence on average density as well as pointwise
density. The modified CH2 system (MCH2) does admit peakon solutions in velocity
and average density. We analytically identify the steepening mechanism that
allows the singular solutions to emerge from smooth spatially-confined initial
data. Numerical results for MCH2 are given and compared with the pure CH2 case.
These numerics show that the modification in MCH2 to introduce average density
has little short-time effect on the emergent dynamical properties. However, an
analytical and numerical study of pairwise peakon interactions for MCH2 shows a
new asymptotic feature. Namely, besides the expected soliton scattering
behavior seen in overtaking and head-on peakon collisions, MCH2 also allows the
phase shift of the peakon collision to diverge in certain parameter regimes.Comment: 25 pages, 11 figure
Test of Universality in Anisotropic 3D Ising Model
Chen and Dohm predicted theoretically in 2004 that the widely believed
universality principle is violated in the Ising model on the simple cubic
lattice with more than only six nearest neighbours. Schulte and Drope by Monte
Carlo simulations found such violation, but not in the predicted direction.
Selke and Shchur tested the square lattice. Here we check only this
universality for the susceptibility ratio near the critical point. For this
purpose we study first the standard Ising model on a simple cubic lattice with
six nearest neighbours, then with six nearest and twelve next-nearest
neighbours, and compare the results with the Chen-Dohm lattice of six nearest
neighbours and only half of the twelve next-nearest neighbours. We do not
confirm the violation of universality found by Schulte and Drope in the
susceptibility ratio.Comment: 6 pages including 4 figures, Physica A, in pres
Integrable discretizations of the sine-Gordon equation
The inverse scattering theory for the sine-Gordon equation discretized in
space and both in space and time is considered.Comment: 18 pages, LaTeX2
New Shape Invariant Potentials in Supersymmetric Quantum Mechanics
Quantum mechanical potentials satisfying the property of shape invariance are
well known to be algebraically solvable. Using a scaling ansatz for the change
of parameters, we obtain a large class of new shape invariant potentials which
are reflectionless and possess an infinite number of bound states. They can be
viewed as q-deformations of the single soliton solution corresponding to the
Rosen-Morse potential. Explicit expressions for energy eigenvalues,
eigenfunctions and transmission coefficients are given. Included in our
potentials as a special case is the self-similar potential recently discussed
by Shabat and Spiridonov.Comment: 8pages, Te
An Inverse Scattering Transform for the Lattice Potential KdV Equation
The lattice potential Korteweg-de Vries equation (LKdV) is a partial
difference equation in two independent variables, which possesses many
properties that are analogous to those of the celebrated Korteweg-de Vries
equation. These include discrete soliton solutions, Backlund transformations
and an associated linear problem, called a Lax pair, for which it provides the
compatibility condition. In this paper, we solve the initial value problem for
the LKdV equation through a discrete implementation of the inverse scattering
transform method applied to the Lax pair. The initial value used for the LKdV
equation is assumed to be real and decaying to zero as the absolute value of
the discrete spatial variable approaches large values. An interesting feature
of our approach is the solution of a discrete Gel'fand-Levitan equation.
Moreover, we provide a complete characterization of reflectionless potentials
and show that this leads to the Cauchy matrix form of N-soliton solutions
Two-component {CH} system: Inverse Scattering, Peakons and Geometry
An inverse scattering transform method corresponding to a Riemann-Hilbert
problem is formulated for CH2, the two-component generalization of the
Camassa-Holm (CH) equation. As an illustration of the method, the multi -
soliton solutions corresponding to the reflectionless potentials are
constructed in terms of the scattering data for CH2.Comment: 22 pages, 3 figures, draft, please send comment
The lattice Schwarzian KdV equation and its symmetries
In this paper we present a set of results on the symmetries of the lattice
Schwarzian Korteweg-de Vries (lSKdV) equation. We construct the Lie point
symmetries and, using its associated spectral problem, an infinite sequence of
generalized symmetries and master symmetries. We finally show that we can use
master symmetries of the lSKdV equation to construct non-autonomous
non-integrable generalized symmetries.Comment: 11 pages, no figures. Submitted to Jour. Phys. A, Special Issue SIDE
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