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Physics and mathematics

of nonlinear processes and models

(University "La Sapienza" , Rome, Italy)

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Brief overview of the main scientific results obtained by
Mario Bruschi
 


MB research activity has focussed mainly (but not only) in non-linear dynamics (on its "integrable" side). 


 
* Results in the field of Non-Linear 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 (Non-Abelian Toda Lattice, Volterra-like 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 integro-differential 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 many-body 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 N-body problems in the ordinary 3-dimensional space  and in the plane . All these models, characterized by Newtonian equations of motion, are rotation-invariant, some of them are moreover translation-invariant, some of them are also Hamiltonian. Moreover several integrable many-body problems were obtained, characterized by autonomous rotation-invariant Newtonian equations of motion (“acceleration equal force”) in d-dimensional 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 Fermi-Pasta-Ulam 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, non-linear, symplectic IM (that can be considered as discrete time Hamiltonian dynamical systems) ;
 
_ an extension of the QR algorithm to construct new explicitly solvable non-linear 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 wave-like and particle-like structures, solitonic and non-solitonic 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 Collatz-Ulam Conjecture (possibly useful as  toy models in parallel computing by Cellular Automata). These CA also suggested a nice generalization of the Collatz-Ulam Conjecture itself.
 


* Mathematical results (a great many of them were obtained as a spin-off of the investigation in the field of non-linear 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 finite-dimensional representation of the derivative operator ;
 
_ solution of a peculiar matrix equation which yields an explicit construction of the commutator and anti-commutator operators ;
 
_ a new method to simplify and classify 'Knot Diagrams' (the relevance of knots in the theoretic physics is increasing: Yang-Baxter 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 Korteweg-de 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 so-called "soliton revolution").