The common trend of our research
group is the study of nonlinear phenomena via techniques as much as
possible “exact”.
Hence it can be seen as
originating from the “soliton revolution” in
mathematical physics initiated at the end of the 1960’s with the discovery of
exact techniques to solve certain nonlinear PDEs (KdV, NLS, sineGordon, etc.)
and, in the following decades, of many novel mathematical findings and related
developments in applicative sectors.
The specific research areas on
which various members of our group have been and are focussing include both the
identification and investigation of new, nontrivial, integrable and solvable
nonlinear evolution equations and dynamical systems (in continuous and discrete
settings), and the application of these results and related techniques in
specific applicative contexts.
For instance, in nonlinear
optics, to explore the experimental verification of new behaviours of
electromagnetic waves, such as those associated with the “boomeronic”
phenomenology; in fluid dynamics, to understand the universal aspects of
such fundamental phenomena as the breaking of waves; in chemistry (or
ecology), to conjecture the possible existence of selfsustaining reactions
(or evolutions) which are not only periodic but even isochronous (i. e.,
periodic with a fixed period).
Other research lines point to a
new understanding of the transition from regular to irregular behaviours of
dynamical systems in terms of travel over Riemann surfaces; the manyfaceted
study of isochronous systems, also leading to certain mathematical findings such
as Diophantine relations; investigations around the notion of the “degree of
integrability” of various systems (again, both in continuous and discrete
settings), relevant to promote the theoretical understanding of nonlinear
evolutions and also to illuminate the accuracy of different approaches to
explore them via numerical simulations.
Indeed one of the longterm goals
of our studies is to arrive at a more refined understanding of the difference
among integrable and chaotic behaviours than the simple dichotomic distinction
among these two phenomenologies as presently identified via definitions which
refer to the behaviour over infinite time of the system (or specific evolution)
under consideration; whereas, as some of our findings suggest (even in the
realistic context of the investigation of the most general manybody problem),
it would seem more appropriate to introduce a distinction among these behaviours
that refers to a specific, finite time scale.
