1,788 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 multiple scales approach to maximal superintegrability
In this paper we present a simple, algorithmic test to establish if a
Hamiltonian system is maximally superintegrable or not. This test is based on a
very simple corollary of a theorem due to Nekhoroshev and on a perturbative
technique called multiple scales method. If the outcome is positive, this test
can be used to suggest maximal superintegrability, whereas when the outcome is
negative it can be used to disprove it. This method can be regarded as a finite
dimensional analog of the multiple scales method as a way to produce soliton
equations. We use this technique to show that the real counterpart of a
mechanical system found by Jules Drach in 1935 is, in general, not maximally
superintegrable. We give some hints on how this approach could be applied to
classify maximally superintegrable systems by presenting a direct proof of the
well-known Bertrand's theorem.Comment: 30 pages, 4 figur
The Calogero-Fran\c{c}oise integrable system: algebraic geometry, Higgs fields, and the inverse problem
We review the Calogero-Fran\c{c}oise integrable system, which is a
generalization of the Camassa-Holm system. We express solutions as (twisted)
Higgs bundles, in the sense of Hitchin, over the projective line. We use this
point of view to (a) establish a general answer to the question of
linearization of isospectral flow and (b) demonstrate, in the case of two
particles, the dynamical meaning of the theta divisor of the spectral curve in
terms of mechanical collisions. Lastly, we outline the solution to the inverse
problem for CF flows using Stieltjes' continued fractions.Comment: 22 pages, 2 figure
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