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# Federico Zullo

INdAM-COFUND Outgoing fellow between 2012-01-10 and 2014-01-09

### Project information

#### Title

BTceq: Backlund Transformations for Classical and Quantum Integrable Systems

#### Outgoing host organisation Return host organisation

University of Kent Università degli Studi Roma Tre

Cornwallis Building Via della Vasca Navale 84

CT2 7NF Canterbury, United Kingdom 00146 Roma, Italy

http://www.kent.ac.uk/ http://www.fis.uniroma3.it/

#### Abstract

The research project is conceived on the potential applications and properties of the theory of Bäcklund transformations to non-linear evolution equations, in particular to finite-dimensional integrable systems, both quantum and classical. This line of research covers a broad spectrum of aspects related to integrable discretizations and exact-time discretizations, separation of variables and factorization of eigenfunctions.

For the classical case the main goal would be to enlighten the interplay between the Bäcklund transformations theory and the classical results of Hamiltonian mechanics and separation of variables, elucidating the role that these transformations could play as a tool to obtain exact time-discretization of the underlying continuous integrable classical model; here the adjective exact means the identity among the discrete trajectories described by the map defined by the transformations and the continuous trajectories defined by the integrable model.

A second aspect would be to clarify to what extent the above constructions can be, *mutatis mutandis*, transferred to the infinite dimensional case, i.e. to Bäcklund transformations for partial differential or differential-difference equations.

These findings could give then the possibility to better understand the noticed connections among the Baxter Q-operator, the Baxter equation and quantum Bäcklund transformations. The link with the classical case is represented by the integral kernel of the Q-operator, which in the semi-classical limit is exactly the generating function of the canonical transformations defined by the Bäcklund transformations.

#### Year 1 Report

The report on the first year of the project can be downloaded from this link

### Fellow information

#### Research interests

Current research interests are twofold:

- Bäcklund Transformations Theory, Hamiltonian and

Dissipative Non-Linear Evolution Equations, Soliton Theory, Integrable

Systems, their geometric and algebraic properties and their integrable

discretizations. - Ecological modelling, population dynamics and

sustainable development, in particular with application to the concept

of “exergy”

#### Previous positions, awards

*January 2011*: Phd in Mathematical Physics at the Doctoral School in Mathematics and Physics, Physics Department of Roma TRE University*September 2005*: MSc (Laurea Magistrale) in Physics cum Laude, Università “Roma Tre”*2005*: awarded with Premio Galluzzi per la fisica 2005 (academic prize for high achievements in graduate studies)*July 2003*: BSc (Laurea) in Physics cum Laude, Università “Roma Tre”

#### Publications, preprints, other works

- O. Ragnisco, F. Zullo,
*Continuous and Discrete (Classical) Heisenberg Spin Chain revised*, SIGMA (Symmetry, Integrability and Geometry:

Methods and Applications) 3 (2007), 033 - O. Ragnisco, F. Zullo,
*Bäcklund Transformations for the Trigonometric*, SIGMA (Symmetry, Integrability and Geometry:

Gaudin Magnet: first results

Methods and Applications) 6 (2010), 012, 6 pages - E. Sciubba, F. Zullo,
*Exergy based population dynamics: a thermodynamic*, 2011. Journal of Industrial Ecology, 15, Issue 2,

view of the sustainability concept

pp. 172â184, April 2011 - E. Sciubba, F. Zullo,
*Is Sustainability a Thermodynamic concept?*, International Journal of Exergy 2011 – 8, No.1

Â pp.Â 68 â 85 - O. Ragnisco, F. Zullo,
*Bäcklund transformations as exact integrable*, Journal of Physics A: Mathematical and

time-discretizations for the trigonometric Gaudin model

Theoretical. 43, 434029, 2010 - O. Ragnisco, F. Zullo,
*Bäcklund Transformation for the Kirchhoff Top*, SIGMA (Symmetry, Integrability and Geometry:

Methods and Applications), 7 (2011), 001, 13 pages - F. Zullo,
*Bäcklund Transformation for the elliptic Gaudin*, J. Math. Phys. 52, 073507 (2011)

model and a Clebsch system - O. Ragnisco, F. Zullo,
*Quantum Bäcklund Transformation: some ideas and examples*, contribution to Solitons in 1+1 and 2+1

dimensions, DS, KP and all that, conference in honor of the 70th

birthday of Marco Boiti and Flora Pempinelli. Lecce, 2011 September

13-14. To appear in Theoretical and Mathematical Physics - E. Sciubba, F. Zullo,
*Stability and limit cycles in an exergy-based*, contribution to Proceedings of ECOS 2012, 25th

model of population dynamics

International Conference on Efficiency, Cost, Optimization, Simulation

and Environmental Impact of Energy Systems June 26-29, 2012, Perugia,

Italy

#### Conferences, schools, other events

*Workshop Nonlinear Waves and Integrable Systems*

Roma Tre University, Department of Physics, Roma,

Italy, February 2009*Ravello Summer School in Mathematical Physics,*

XXXIV edition

Ravello, Italy, September 2009*Trasformazioni di Bäcklund per il modello*

trigonometrico di Gaudin (Bäcklund Transformations for the

trigonometric Gaudin model)

Talk delivered at*Assemblea Scientifica G.N.F.M.*

Montecatini Terme, Italy, October 2009- Plenary talk
*“An exact time discretization*

for the Kirchhoff top via Bäcklund transformation for the

trigonometric Gaudin Model”

at*XIXth International Colloquium on*

Integrable Systems and Quantum Symmetries

Prague, Czech Republic, June 2010 *London Mathematical Society and EPSRC Short*

Instructional Course on Classical & Quantum Integrable Models

Canterbury, University of Kent, United Kingdom, July 2010*Bäcklund transformations for the elliptic Gaudin*

magnet

Talk delivered at*Department of Mathematics and Applications,*

University of Milano-Bicocca

Milano, Italy, October 2010*Bäcklund transformation and time-discretizations*

for the elliptic Gaudin Model

workshop*Discrete Integrable Systems*

Lorentz Center, Leiden, Holland, 18-22 July 2011*Separation of variables, Bäcklund transformations*

and “spectrality property” for finite dimensional integrable

systems

conference*Finite Dimensional Integrable*

Systems in Geometry and Mathematical Physics

Jena, Germany, 26-29 July 2011*Exact time discretisations via Bäcklund*

transformations: general ideas and examples

Talk delivered at*Bi-Hamiltonian Systems and All That, Conference in*

honour of Franco Magri’s 65th birthday

Dipartimento di Matematica e Applicazioni,

UniversitÃ di Milano-Bicocca & Dipartimento di Ingegneria

dell’Informazione e Metodi Matematici, UniversitÃ di Bergamo, 27 September-1 October 2011- Invited talk
*“Exergy-driven population*

dynamics”

at*Dipartimento di Ingegneria Elettronica,*

Dipartimento di Fisica & INFN â Sezione di Roma Tre, Roma Tre

University

Roma, Italy, 15 December 2011 *British Mathematical Colloquium & Mathematical*

Physical Workshop

University of Kent, Canterbury, Kent, 16-19 April 2012*Euroscience Open Forum & Marie Curie Actions*

Conference

Dublin, Ireland, 10-15 July 2012- Invited instructor at
*Summer School of*

Thermodynamics

Anzio, Italy, 11-23 June 2012 *Assemblea Scientifica GNFM 2012*

Montecatini Terme, Italy, 4-6 October 2012