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Physics and mathematics

of nonlinear processes and models

(University "La Sapienza" , Rome, Italy)

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Matteo Sommacal

Matteo Sommacal's main scientific interests span over many topics in the field of Mathematical-Physics and can be essentially collected in 7 main research lines, each of them corresponding to several fruitful, long-standing and well-established scientific collaborations.

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[1] Isochronicity and the transition from ordered to disordered motions for dynamical systems

Short Description: Study and classification of integrable and non-integrable many-body problems featuring periodic solutions. Applications to periodic (isochronous) natural phenomena (e.g. periodic chemical reactions). Study of the transitions from ordered to disordered motions for dynamical systems, explained as travels on Riemann surfaces. Analysis of the mechanisms for the onset of irregular (chaotic) motions in a deterministic context and definition of an alternative route to explaining chaotic motions via the analytic structure (branching) of the solutions in the complex time plane.

Keywords: integrable and super-integrable systems; dynamical systems; Hamiltonian systems; isochronicity; singularities in complex time; transitions to chaos; Lyapunov exponents; PainlevÚ and poly-PainlevÚ analysis; Psi-series; Riemann surfaces; complex analysis; monodromy group; Fuchsian equations; continued fractions; numerical integration of ordinary differential equations; periodic chemical reactions.

Collaborations: Francesco Calogero and Paolo Maria Santini, UniversitÓ degli Studi di Roma "La Sapienza" (Italy) and David Gomez-Ullate, Universidad Complutense de Madrid (Spain); other results where obtained in collaboration with Jean-Pierre Franšoise, UniversitÚ P. et M. Curie, Paris VI, Paris (France); Andrew Hone, University of Kent (UK); Franšois Leyvraz, Universidad de los Andes, BogotÓ (Columbia); Piotr Grinevich, Landau Institut, Moscow (Russia).

Period: 2002 - Present days.

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[2] Continuous elastic rods as models for polymeric chains and protein folding

Short Description: Analytical and numerical study of the dynamical Kirchhoff equations for an elastic rod (a system of nine coupled nonlinear PDEs in 1+1 dimensions) as a model for polymeric chains and protein folding. Full classification of all the circular helical (stable and unstable) con gurations for the static problem. Analysis of the time evolution of a strainable rod.

Keywords: continuous media; isotropic and anisotropic elasticity; Cosserat continuum; De Saint Venant's theory of beams; coarse-grained structures; mechanical properties and constitutive equations of a thin elastic rod; Young and shear moduli; Kirchhoff equations; numerical integration of systems of PDEs; perturbative and asymptotic analysis; alpha helices and beta sheets; polymeric chains; protein folding.

Collaborations: Mark J. Ablowitz, Colorado University at Boulder (CO, USA); Vincenzo Barone, Scuola Normale

Superiore di Pisa (Italy); Mario Argeri, UniversitÓ degli Studi di Napoli Federico II (Italy); Silvana De Lillo and Gaia Lupo, UniversitÓ di Perugia (Italy).

Period: 2007 - Present days.

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[3] Moving magnetic droplet solitons

Short Description: Landau-Lifshitz equation in two spatial dimensions as a model for spin-wave excitation using a point contact in a thin ferromagnetic film.

Keywords: Landau-Lifshitz equation; spin-wave excitation; localized solutions; numerical integration of PDEs.

Collaborations: Mark Hoefer, North Carolina State University, Raleigh (NC, USA).

Period: 2009 - Present days.

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[4] The onset and the interaction of dispersive and dissipative shock waves

Short Description: Study of the onset and of the interaction of dispersive and dissipative shock waves as solutions of integrable and nonintegrable nonlinear evolution equations, starting from Kamchatnov theory on the approximation of Whitham modulation equations to describe oscillatory solutions in dispersive means.

Keywords: inverse scattering transform; shock waves; Whitham equations; elliptic integrals; perturbative and asymptotic analysis; Burgers equation; KdV equation; NLS equation; KdV-Burgers equation; Bose-Einstein condensates; numerical integration of PDEs.

Collaborations: Silvana De Lillo and Gaia Lupo, UniversitÓ degli Studi di Perugia (Italy); Mark J. Ablowitz, Colorado University at Boulder (CO, USA); Mark Hoefer, North Carolina State University, Raleigh (NC, USA).

Period: 2007 - Present days.

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[5] Evolutionary integrable models in multidimension

Short Description: Formulation of new solvable nonlinear (systems of) evolution PDEs in multidimensional space, involving rational, trigonometric and elliptic functions of the independent variables.

Keywords: nonlinear evolution PDEs in multidimensions; solvable PDEs; NLS-like equations; nonlinear Klein-Gordon-like equations; elliptic functions; isochronicity; numerical integration of PDEs.

Collaborations: Francesco Calogero, UniversitÓ degli Studi di Roma "La Sapienza" (Italy) and Jean-Pierre Franšoise, UniversitÚ P. et M. Curie, Paris VI, Paris (France).

Period: 2006 - 2007.

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[6] Initial conditions/boundary value problems

Short description: (1) Nonlinear heat conduction problems for Storm materials. (2) The Neumann problem on the semi-line for the Burgers equation. (3) A nonlinear model of sulphation phenomena in calcium carbonate stones. Keywords: Dirichlet, Neumann and Robin problems; moving boundaries; Hopf-Cole transformation; hodographic transformations; numerical integration of PDEs.

Collaborations: Silvana De Lillo and Gaia Lupo, UniversitÓ di Perugia (Italy).

Period: 2006 - 2007.

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[7] Combinatoric and geometrical methods for Archaeology

Short Description: (1) Development and numerical implementation of seriation techniques to derive the relative chronology of archaeological data. (2) Polynomial interpolation techinques (cubic spline) to define cultural and political boundaries from topographic archaeological data.

Keywords:relative chronologies; Ihm algorithm; the traveling salesman's problem; optimization problems; Voronoi diagrams; Hamiltonian paths; labeled graphs; cubic splines.

Collaborations: European Proto-History Chair, Paola Piana Agostinetti, UniversitÓ degli Studi "La Sapienza", Roma (Italy).

Period: 2004 - Present days.