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Physics and mathematics

of nonlinear processes and models

(University "La Sapienza" , Rome, Italy)

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  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, sine-Gordon, 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 self-sustaining 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 many-faceted 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 long-term 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 many-body problem), it would seem more appropriate to introduce a distinction among these behaviours that refers to a specific, finite time scale.