182 research outputs found
A heat transfer with a source: the complete set of invariant difference schemes
In this letter we present the set of invariant difference equations and
meshes which preserve the Lie group symmetries of the equation
u_{t}=(K(u)u_{x})_{x}+Q(u). All special cases of K(u) and Q(u) that extend the
symmetry group admitted by the differential equation are considered. This paper
completes the paper [J. Phys. A: Math. Gen. 30, no. 23 (1997) 8139-8155], where
a few invariant models for heat transfer equations were presented.Comment: arxiv version is already officia
Gas flow with straight transition line
An investigation was conducted on the limiting case of a gas flow when the constant pressure in the surrounding medium is exactly equal to the critical pressure for the given initial state of the gas
Symmetry-preserving discrete schemes for some heat transfer equations
Lie group analysis of differential equations is a generally recognized
method, which provides invariant solutions, integrability, conservation laws
etc. In this paper we present three characteristic examples of the construction
of invariant difference equations and meshes, where the original continuous
symmetries are preserved in discrete models. Conservation of symmetries in
difference modeling helps to retain qualitative properties of the differential
equations in their difference counterparts.Comment: 21 pages, 4 ps figure
Symmetry group analysis of an ideal plastic flow
In this paper, we study the Lie point symmetry group of a system describing
an ideal plastic plane flow in two dimensions in order to find analytical
solutions. The infinitesimal generators that span the Lie algebra for this
system are obtained. We completely classify the subalgebras of up to
codimension two in conjugacy classes under the action of the symmetry group.
Based on invariant forms, we use Ansatzes to compute symmetry reductions in
such a way that the obtained solutions cover simultaneously many invariant and
partially invariant solutions. We calculate solutions of the algebraic,
trigonometric, inverse trigonometric and elliptic type. Some solutions
depending on one or two arbitrary functions of one variable have also been
found. In some cases, the shape of a potentially feasible extrusion die
corresponding to the solution is deduced. These tools could be used to thin,
curve, undulate or shape a ring in an ideal plastic material
Optical Lattice Polarization Effects on Hyperpolarizability of Atomic Clock Transitions
The light-induced frequency shift due to the hyperpolarizability (i.e. terms
of second-order in intensity) is studied for a forbidden optical transition,
=0=0. A simple universal dependence on the field ellipticity is
obtained. This result allows minimization of the second-order light shift with
respect to the field polarization for optical lattices operating at a magic
wavelength (at which the first-order shift vanishes). We show the possibility
for the existence of a magic elliptical polarization, for which the
second-order frequency shift vanishes. The optimal polarization of the lattice
field can be either linear, circular or magic elliptical. The obtained results
could improve the accuracy of lattice-based atomic clocks.Comment: 4 pages, RevTeX4, 2 eps fig
Group properties and invariant solutions of a sixth-order thin film equation in viscous fluid
Using group theoretical methods, we analyze the generalization of a
one-dimensional sixth-order thin film equation which arises in considering the
motion of a thin film of viscous fluid driven by an overlying elastic plate.
The most general Lie group classification of point symmetries, its Lie algebra,
and the equivalence group are obtained. Similar reductions are performed and
invariant solutions are constructed. It is found that some similarity solutions
are of great physical interest such as sink and source solutions,
travelling-wave solutions, waiting-time solutions, and blow-up solutions.Comment: 8 page
Symmetry reductions of a particular set of equations of associativity in twodimensional topological field theory
The WDVV equations of associativity arising in twodimensional topological
field theory can be represented, in the simplest nontrivial case, by a single
third order equation of the Monge-Ampe`re type. By investigating its Lie point
symmetries, we reduce it to various nonlinear ordinary differential equations,
and we obtain several new explicit solutions.Comment: 10 pages, Latex, to appear in J. Phys. A: Math. Gen. 200
Lie symmetry analysis and exact solutions of the quasi-geostrophic two-layer problem
The quasi-geostrophic two-layer model is of superior interest in dynamic
meteorology since it is one of the easiest ways to study baroclinic processes
in geophysical fluid dynamics. The complete set of point symmetries of the
two-layer equations is determined. An optimal set of one- and two-dimensional
inequivalent subalgebras of the maximal Lie invariance algebra is constructed.
On the basis of these subalgebras we exhaustively carry out group-invariant
reduction and compute various classes of exact solutions. Where possible,
reference to the physical meaning of the exact solutions is given. In
particular, the well-known baroclinic Rossby wave solutions in the two-layer
model are rediscovered.Comment: Extended version, 24 pages, 1 figur
Ultrastable Optical Clock with Neutral Atoms in an Engineered Light Shift Trap
An ultrastable optical clock based on neutral atoms trapped in an optical
lattice is proposed. Complete control over the light shift is achieved by
employing the transition of
atoms as a "clock transition". Calculations of ac multipole polarizabilities
and dipole hyperpolarizabilities for the clock transition indicate that the
contribution of the higher-order light shifts can be reduced to less than 1
mHz, allowing for a projected accuracy of better than .Comment: 4 pages, 2 figures, accepted for publication in Phys. Rev. Let
Symmetry Analysis of Barotropic Potential Vorticity Equation
Recently F. Huang [Commun. Theor. Phys. V.42 (2004) 903] and X. Tang and P.K.
Shukla [Commun. Theor. Phys. V.49 (2008) 229] investigated symmetry properties
of the barotropic potential vorticity equation without forcing and dissipation
on the beta-plane. This equation is governed by two dimensionless parameters,
and , representing the ratio of the characteristic length scale to
the Rossby radius of deformation and the variation of earth' angular rotation,
respectively. In the present paper it is shown that in the case there
exists a well-defined point transformation to set . The
classification of one- and two-dimensional Lie subalgebras of the Lie symmetry
algebra of the potential vorticity equation is given for the parameter
combination and . Based upon this classification, distinct
classes of group-invariant solutions is obtained and extended to the case
.Comment: 6 pages, release version, added reference for section
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