48 research outputs found
Symbolic Software for the Painleve Test of Nonlinear Ordinary and Partial Differential Equations
The automation of the traditional Painleve test in Mathematica is discussed.
The package PainleveTest.m allows for the testing of polynomial systems of
ordinary and partial differential equations which may be parameterized by
arbitrary functions (or constants). Except where limited by memory, there is no
restriction on the number of independent or dependent variables. The package is
quite robust in determining all the possible dominant behaviors of the Laurent
series solutions of the differential equation. The omission of valid dominant
behaviors is a common problem in many implementations of the Painleve test, and
these omissions often lead to erroneous results. Finally, our package is
compared with the other available implementations of the Painleve test.Comment: Published in the Journal of Nonlinear Mathematical Physics
(http://www.sm.luth.se/math/JNMP/), vol. 13(1), pp. 90-110 (Feb. 2006). The
software can be downloaded at either http://www.douglasbaldwin.com or
http://www.mines.edu/fs_home/wherema
A list of all integrable 2D homogeneous polynomial potentials with a polynomial integral of order at most 4 in the momenta
We searched integrable 2D homogeneous polynomial potential with a polynomial
first integral by using the so-called direct method of searching for first
integrals. We proved that there exist no polynomial first integrals which are
genuinely cubic or quartic in the momenta if the degree of homogeneous
polynomial potentials is greater than 4.Comment: 22 pages, no figures, to appear in J. Phys. A: Math. Ge
Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom
We consider natural complex Hamiltonian systems with degrees of freedom
given by a Hamiltonian function which is a sum of the standard kinetic energy
and a homogeneous polynomial potential of degree . The well known
Morales-Ramis theorem gives the strongest known necessary conditions for the
Liouville integrability of such systems. It states that for each there
exists an explicitly known infinite set \scM_k\subset\Q such that if the
system is integrable, then all eigenvalues of the Hessian matrix V''(\vd)
calculated at a non-zero \vd\in\C^n satisfying V'(\vd)=\vd, belong to
\scM_k. The aim of this paper is, among others, to sharpen this result. Under
certain genericity assumption concerning we prove the following fact. For
each and there exists a finite set \scI_{n,k}\subset\scM_k such that
if the system is integrable, then all eigenvalues of the Hessian matrix
V''(\vd) belong to \scI_{n,k}. We give an algorithm which allows to find
sets \scI_{n,k}. We applied this results for the case and we found
all integrable potentials satisfying the genericity assumption. Among them
several are new and they are integrable in a highly non-trivial way. We found
three potentials for which the additional first integrals are of degree 4 and 6
with respect to the momenta.Comment: 54 pages, 1 figur
Perturbed Three Vortex Dynamics
It is well known that the dynamics of three point vortices moving in an ideal
fluid in the plane can be expressed in Hamiltonian form, where the resulting
equations of motion are completely integrable in the sense of Liouville and
Arnold. The focus of this investigation is on the persistence of regular
behavior (especially periodic motion) associated to completely integrable
systems for certain (admissible) kinds of Hamiltonian perturbations of the
three vortex system in a plane. After a brief survey of the dynamics of the
integrable planar three vortex system, it is shown that the admissible class of
perturbed systems is broad enough to include three vortices in a half-plane,
three coaxial slender vortex rings in three-space, and `restricted' four vortex
dynamics in a plane. Included are two basic categories of results for
admissible perturbations: (i) general theorems for the persistence of invariant
tori and periodic orbits using Kolmogorov-Arnold-Moser and Poincare-Birkhoff
type arguments; and (ii) more specific and quantitative conclusions of a
classical perturbation theory nature guaranteeing the existence of periodic
orbits of the perturbed system close to cycles of the unperturbed system, which
occur in abundance near centers. In addition, several numerical simulations are
provided to illustrate the validity of the theorems as well as indicating their
limitations as manifested by transitions to chaotic dynamics.Comment: 26 pages, 9 figures, submitted to the Journal of Mathematical Physic
Non integrability of a self-gravitating Riemann liquid ellipsoid
We prove that the motion of a triaxial Riemann ellipsoid of homogeneous
liquid without angular momentum does not possess an additional first integral
which is meromorphic in position, impulsions, and the elliptic functions which
appear in the potential, and thus is not integrable. We prove moreover that
this system is not integrable even on a fixed energy level hypersurface.Comment: 14 pages, 8 reference
Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: a Review and Extensions of Tests for the Painlev\'e Property
The integrability (solvability via an associated single-valued linear
problem) of a differential equation is closely related to the singularity
structure of its solutions. In particular, there is strong evidence that all
integrable equations have the Painlev\'e property, that is, all solutions are
single-valued around all movable singularities. In this expository article, we
review methods for analysing such singularity structure. In particular, we
describe well known techniques of nonlinear regular-singular-type analysis,
i.e. the Painlev\'e tests for ordinary and partial differential equations. Then
we discuss methods of obtaining sufficiency conditions for the Painlev\'e
property. Recently, extensions of \textit{irregular} singularity analysis to
nonlinear equations have been achieved. Also, new asymptotic limits of
differential equations preserving the Painlev\'e property have been found. We
discuss these also.Comment: 40 pages in LaTeX2e. To appear in the Proceedings of the CIMPA Summer
School on "Nonlinear Systems," Pondicherry, India, January 1996, (eds) B.
Grammaticos and K. Tamizhman
Systems of Hess-Appel'rot type
We construct higher-dimensional generalizations of the classical
Hess-Appel'rot rigid body system. We give a Lax pair with a spectral parameter
leading to an algebro-geometric integration of this new class of systems, which
is closely related to the integration of the Lagrange bitop performed by us
recently and uses Mumford relation for theta divisors of double unramified
coverings. Based on the basic properties satisfied by such a class of systems
related to bi-Poisson structure, quasi-homogeneity, and conditions on the
Kowalevski exponents, we suggest an axiomatic approach leading to what we call
the "class of systems of Hess-Appel'rot type".Comment: 40 pages. Comm. Math. Phys. (to appear
Quasi-analytical solutions for APSIDAL motion in the three-body problem: Sun—minor planet—Jupiter
This paper deals with the effect of a third body on the apsidal motion of two bodies. The specific case involves a third body-planet Jupiter and the apsidal line motion of a minor planet that orbits the Sun and has its apsidal line go through the major axis of an ellipse. The third body (Jupiter) which satisfies the Langrangian solution will affect the apsidal line motion and therefore affects the ascending and descending motions of the minor planet. In this case no analytical solutions can be obtained, and therefore specific assumptions are made along with numerical solutions. For convenience, we adopt the Lagrangian solution in the three-body problem and obtain quasi-analytical results, which are used to evaluate the effect of the planet on the d Omega/dt (Omega ascending node) of each minor planet. This method is beneficial for improving our knowledge of the orbital elements of the asteroids, and perhaps even much smaller effects such as the effects of the planets on the interplanetary dust complex. Information on the latter may be provided by using this method to investigate Jupiter\u27s effect on the inclination of the symmetry surface of the zodiacal dust cloud
Motion of Three Vortices near Collapse
A system of three point vortices in an unbounded plane has a special family
of self-similarly contracting or expanding solutions: during the motion, vortex
triangle remains similar to the original one, while its area decreases (grows)
at a constant rate. A contracting configuration brings three vortices to a
single point in a finite time; this phenomenon known as vortex collapse is of
principal importance for many-vortex systems. Dynamics of close-to-collapse
vortex configurations depends on the way the collapse conditions are violated.
Using an effective potential representation, a detailed quantitative analysis
of all the different types of near-collapse dynamics is performed when two of
the vortices are identical. We discuss time and length scales, emerging in the
problem, and their behavior as the initial vortex triangle is approaching to an
exact collapse configuration. Different types of critical behaviors, such as
logarithmic or power-law divergences are exhibited, which emphasizes the
importance of the way the collapse is approached. Period asymptotics for all
singular cases are presented as functions of the initial vortices
configurations. Special features of passive particle mixing by a near-collapse
flows are illustrated numerically.Comment: 45 pages, 22 figures Last version of the paper with all update
Jets, Stickiness and Anomalous Transport
Dynamical and statistical properties of the vortex and passive particle
advection in chaotic flows generated by four and sixteen point vortices are
investigated. General transport properties of these flows are found anomalous
and exhibit a superdiffusive behavior with typical second moment exponent (\mu
\sim 1.75). The origin of this anomaly is traced back to the presence of
coherent structures within the flow, the vortex cores and the region far from
where vortices are located. In the vicinity of these regions stickiness is
observed and the motion of tracers is quasi-ballistic. The chaotic nature of
the underlying flow dictates the choice for thorough analysis of transport
properties. Passive tracer motion is analyzed by measuring the mutual relative
evolution of two nearby tracers. Some tracers travel in each other vicinity for
relatively large times. This is related to an hidden order for the tracers
which we call jets. Jets are localized and found in sticky regions. Their
structure is analyzed and found to be formed of a nested sets of jets within
jets. The analysis of the jet trapping time statistics shows a quantitative
agreement with the observed transport exponent.Comment: 17 pages, 17 figure