
Brief overview of the main scientific results obtained by
Mario Bruschi
MB research activity has focussed mainly (but not only) in
nonlinear dynamics (on its "integrable" side).
* Results in the field of NonLinear Evolution Equations (continuous
(NLEE) and discrete (DNLEE)):
_ introduction of new integrable NLEE, investigation of
their peculiar solitonic solutions and analysis of their main
mathematical properties (Backlund Transformations, Lax Pairs,
Nonlinear Superposition, etc.);
_ investigation of integrable DNLEE: discrete versions of
continuous NLEE , introduction of the direct and inverse Spectral
Problem for the Discrete Matrix Schroedinger Operator and analysis
of the DNLEE associated with isospectral and non isospectral
deformations of this operator (NonAbelian Toda Lattice,
Volterralike Equations, etc.) ;
_ Hamiltonian structure of certain NLEE and DNLEE, their
integrability through the construction of an appropriate infinite
set of constants of motion ;
_ a novel, simple and powerful technique (based on the algebra of
the differential operators) to construct the Hierarchies of NLEE {DNLEE}
associated with the isospectral and non isospectral deformations of
a given linear integrodifferential operator {finite difference
operator}. This technique allows also to construct explicitly and
simultaneously the Lax Pair for each equation in the hierarchy ( and
thus it provides the evolution of the eigenfunctions of the
associated linear operator). Moreover it allows to recover the
hierarchy of the associated Darboux and Backlund Transformations and
, through the explicit construction of the Recurrence Operator,
strongly suggests the possible underlying Hamiltonian structure for
the hierarchy itself.
* Results in the field of solvable and/or integrable manybody
dynamical systems
_ introduction and investigation of new integrable classic (and
quantum) dynamical systems in one dimension (N interacting particles
on the line, Hamiltonian systems with N degrees of freedom). Some of
these dynamical systems are also "relativistic" (invariant under the
Poincarč Group) ;
_ introduction of a great number of solvable and/or integrable
and/or linearizable Nbody problems in the ordinary 3dimensional
space and in the plane . All these models, characterized by
Newtonian equations of motion, are rotationinvariant, some of them
are moreover translationinvariant, some of them are also
Hamiltonian. Moreover several integrable manybody problems were
obtained, characterized by autonomous rotationinvariant Newtonian
equations of motion (“acceleration equal force”) in ddimensional
space (d=1.2.3...) with velocity independent forces linear and cubic
in the particle coordinates. These systems are generally
interpretable as Hamiltonian assemblies of quadratic and quartic
oscillators . The investigation of quartic oscillators is, via the
FermiPastaUlam numerical experiment, at the origin of the
revolution that has occurred, over the last three/four decades, in
the understanding of the behavior and relevance of integrable
systems. It has moreover an obvious and ubiquitous applicative
interest, inasmuch as it generally provides the first nonlinear (“unharmonic”)
correction to the behavior of linear (“harmonic”) oscillators, whose
physical relevance is of course universal. All these findings are
obtained starting from evolution equations for matrices amenable to
exact treatments, through convenient and compatible parametrizations
of the evolving matrices in terms of vectors .
* Results in the field of Discrete Time Dynamical Systems:
Iterated Maps (IM) and Cellular Automata (CA)
_ a method to construct, starting from known integrable
Hamiltonian hierarchies of NLEE, new hierarchies of integrable,
nonlinear, symplectic IM (that can be considered as discrete time
Hamiltonian dynamical systems) ;
_ an extension of the QR algorithm to construct new explicitly
solvable nonlinear IM ;
_ an explicit determinantal solution of the Logistic Map (which
is considered a prototype of the "chaotic" phenomenology) ;
_ extension of the 'standard' mathematical techniques, developed to
integrate the NLEE, to introduce and investigate new hierarchies of
CA in one, two and three dimensions. These CA exhibit many
interesting features: an infinite set of conservation laws, time
reversal invariance, coherent moving wavelike and particlelike
structures, solitonic and nonsolitonic interactions, gluing
phenomena, etc. Due to this rich dynamics, they are suitable to
model real phenomena in Physics, Biology, Meteorology, etc.;
_ introduction of CA which "compute" the CollatzUlam Conjecture (possibly
useful as toy models in parallel computing by Cellular Automata).
These CA also suggested a nice generalization of the CollatzUlam
Conjecture itself.
* Mathematical results (a great many of them were obtained as a
spinoff of the investigation in the field of nonlinear dynamics)
_ new properties of the zeros of the "classical" polynomials and of
the Bessel functions, elegant "diophantine" spectral properties of
NxN matrices (suitable for teaching purposes and numerical computer
tests), analytic general solution of new interesting functional
equations ;
_ a new simple and very powerful method for numerical computation of
the eigenvalues and the eigenfuctions of linear differential
operators (even in more than one dimension): this method is based
upon a simple, suitable finitedimensional representation of the
derivative operator ;
_ solution of a peculiar matrix equation which yields an explicit
construction of the commutator and anticommutator operators ;
_ a new method to simplify and classify 'Knot Diagrams' (the
relevance of knots in the theoretic physics is increasing:
YangBaxter models , strings, etc. );
_ identification of certain polynomials allowing Diophantine
factorizations, including some
polynomials belonging to the standard families of orthogonal polynomial
classified according to the Askey scheme;
_ new Diophantine properties related to the integrable hierarchy of
nonlinear PDEs
associated with the Kortewegde Vries 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) that followed the discovery
at the end
of the 1960's of the integrable character of the KdV equation (the
socalled "soliton revolution").

