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Scientific Report 2007-2009
Theoretical physics
T23. Discrete integrable dynamical systems and Diophantine
relations associated with certain polynomial classes
The main idea underlying the various research lines
pursued under this heading originates from the following
observation. If one knows a nonlinear dynamical system
with an arbitrary number N of degrees of freedom
that is isochronous — namely, that in an open region of
its phase space features solutions all of which are completely
periodic (i. e., periodic in all their degrees of
freedom) with a fixed period (independent of the initial
data, provided it falls within the isochrony region)—and
if an equilibrium solution of this dynamical system can
be explicitly found in the isochrony region, then standard
linearization of the equations of motion of the system in
the immediate neighborhood of this equilibrium configuration
yields an N × N matrix whose eigenvalues must
all be integer multiples of a common factor—because
these eigenvalues yield the frequencies of the oscillations
of the system in the infinitesimal neighborhood of its
equilibrium, and if the system is isochronous, all these
N frequencies must indeed be integer multiples of a common
factor. So, by this roundabout way, one arrives at
a Diophantine finding (i. e., an explicit matrix whose
eigenvalues must all be integers)—a finding which may
become a conjecture if one venture to guess the actual
values of these N integers, or a theorem whenever such
a conjecture can be proven.
This observation also opened a relatively vast area of
research, because—contrary to what one might naively
think—nonlinear isochronous dynamical systems are not
rare, indeed there are techniques to manufacture a lot of
them. For recent papers reporting results of this kind see
[1,2,3]. The paper [3] is particularly remarkable inasmuch
as it yielded new Diophantine properties related
to the integrable hierarchy of nonlinear PDEs associated
with the Korteweg-de Vries (KdV) equation, an item
that has played a pivotal role in the major developments
in theoretical and mathematical physics, and as well in
several fields of pure mathematics, consequential to the
discovery at the end of the 1960’s of the integrable character
of the KdV equation (the so-called ”soliton revolution”).
Moreover, to prove some of the conjectures arrived at
in this manner, we pursued a research line leading to the
identification of certain polynomials allowing Diophantine
factorizations—including some polynomials belonging
to the standard families of orthogonal polynomials
classified according to the Askey scheme (for a recent instance
of such results see [4]). And these developments
have led to the identification of new discrete integrable
systems [4], a finding whose ramifications are still under
investigation. Indeed a new paper of this series, coauthored
by the same group of authors (M. Bruschi, F.
Calogero and R. Droghei), is in preparation.
Overall, the research line tersely outlined above seems
susceptible of significant further developments, which
we plan to pursue in the coming years. We also plan to
promote the insertion of at least some of our findings
in standard compilations of the properties of special
functions, as was the case in the past for some analogous
results: see section 15.823, entitled ”Hermitian matrices
and diophantine relations involving singular functions of
rational angles due to Calogero and Perelomov”, in the
standard compilation of mathematical results originally
due to I. S. Gradshteyn and I. M Rizhik (”Tables of
integrals, series, and products”, fifth edition, edited by
Alan Jeffrey, Academic Press, 1980).
References
1. R. Droghei et al., J. Phys. A42, 454202 (2009).
2. R. Droghei et al., J. Phys A42, 445207 (2009).
3. M. Bruschi et al., J. Math. Phys. 50, 122701 (2009).
4. M. Bruschi et al., Adv. Math. Phys. 2009, 268134 (2009).
Authors
M. Bruschi, F. Calogero 1
http://solitons.altervista.org/
Sapienza Università di Roma 46 Dipartimento di Fisica