933 research outputs found
Finding Liouvillian first integrals of rational ODEs of any order in finite terms
It is known, due to Mordukhai-Boltovski, Ritt, Prelle, Singer, Christopher
and others, that if a given rational ODE has a Liouvillian first integral then
the corresponding integrating factor of the ODE must be of a very special form
of a product of powers and exponents of irreducible polynomials. These results
lead to a partial algorithm for finding Liouvillian first integrals. However,
there are two main complications on the way to obtaining polynomials in the
integrating factor form. First of all, one has to find an upper bound for the
degrees of the polynomials in the product above, an unsolved problem, and then
the set of coefficients for each of the polynomials by the
computationally-intensive method of undetermined parameters. As a result, this
approach was implemented in CAS only for first and relatively simple second
order ODEs. We propose an algebraic method for finding polynomials of the
integrating factors for rational ODEs of any order, based on examination of the
resultants of the polynomials in the numerator and the denominator of the
right-hand side of such equation. If both the numerator and the denominator of
the right-hand side of such ODE are not constants, the method can determine in
finite terms an explicit expression of an integrating factor if the ODE permits
integrating factors of the above mentioned form and then the Liouvillian first
integral. The tests of this procedure based on the proposed method, implemented
in Maple in the case of rational integrating factors, confirm the consistence
and efficiency of the method.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Solutions for the General, Confluent and Biconfluent Heun equations and their connection with Abel equations
In a recent paper, the canonical forms of a new multi-parameter class of Abel
differential equations, so-called AIR, all of whose members can be mapped into
Riccati equations, were shown to be related to the differential equations for
the hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connection
between the AIR canonical forms and the Heun General (GHE), Confluent (CHE) and
Biconfluent (BHE) equations is presented. This connection fixes the value of
one of the Heun parameters, expresses another one in terms of those remaining,
and provides closed form solutions in terms of pFq functions for the resulting
GHE, CHE and BHE, respectively depending on four, three and two irreducible
parameters. This connection also turns evident what is the relation between the
Heun parameters such that the solutions admit Liouvillian form, and suggests a
mechanism for relating linear equations with N and N-1 singularities through
the canonical forms of a non-linear equation of one order less.Comment: Original version submitted to Journal of Physics A: 16 pages, related
to math.GM/0002059 and math-ph/0402040. Revised version according to
referee's comments: 23 pages. Sign corrected (June/17) in formula (79).
Second revised version (July/25): 25 pages. See also
http://lie.uwaterloo.ca/odetools.ht
Kondo model in nonequilibrium: Interplay between voltage, temperature, and crossover from weak to strong coupling
We consider an open quantum system in contact with fermionic metallic
reservoirs in a nonequilibrium setup. For the case of spin, orbital or
potential fluctuations, we present a systematic formulation of real-time
renormalization group at finite temperature, where the complex Fourier variable
of an effective Liouvillian is used as flow parameter. We derive a universal
set of differential equations free of divergencies written as a systematic
power series in terms of the frequency-independent two-point vertex only, and
solve it in different truncation orders by using a universal set of boundary
conditions. We apply the formalism to the description of the weak to strong
coupling crossover of the isotropic spin-1/2 nonequilibrium Kondo model at zero
magnetic field. From the temperature and voltage dependence of the conductance
in different energy regimes we determine various characteristic low-energy
scales and compare their universal ratio to known results. For a fixed finite
bias voltage larger than the Kondo temperature, we find that the
temperature-dependence of the differential conductance exhibits non-monotonic
behavior in the form of a peak structure. We show that the peak position and
peak width scale linearly with the applied voltage over many orders of
magnitude in units of the Kondo temperature. Finally, we compare our
calculations with recent experiments.Comment: 48 pages, 10 figure
A generalization of the S-function method applied to a Duffing-Van der Pol forced oscillator
In [1,2] we have developed a method (we call it the S-function method) that
is successful in treating certain classes of rational second order ordinary
differential equations (rational 2ODEs) that are particularly `resistant' to
canonical Lie methods and to Darbouxian approaches. In this present paper, we
generalize the S-function method making it capable of dealing with a class of
elementary 2ODEs presenting elementary functions. Then, we apply this method to
a Duffing-Van der Pol forced oscillator, obtaining an entire class of first
integrals
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