FP7-PEOPLE-2013-IIF - Marie Curie Action: "International Incoming Fellowships"
Project reference number: 624199
Financed from: FP7-PEOPLE
Period of stay of the principal investigator: from 2014-05-05 to 2015-07-04
Acronym: SPEDIS

Symmetry preserving discretization of integrable, superintegrable and nonintegrable systems.

Project details

The aim of this project is to develop and apply efficient mathematical tools for studying quantum and classical phenomena in a discrete setting.

The motivation is on one hand that on the fundamental level it seems that space-time is discrete, because of the existence of the Planck length and its role e.g. in quantum gravity. On the other hand, even in a continuous world many important phenomena are discrete, such as phenomena occurring in crystals or in molecular or atomic chains. Thus difference equations may be more fundamental than differential ones. Moreover, differential equations often have to be solved numerically and that means that they have to be discretized, i.e. approximated by a difference system.

Our main interest is in models that can be solved exactly because of their symmetry and integrability properties. Of special interest are finite and infinite dimensional integrable and superintegrable models. Integrable systems have as many commuting integrals of motion as degrees of freedom (which may be infinite). Superintegrable systems have more integrals of motion than degrees of freedom and these integrals form interesting non-Abelian algebras. The integrals of motion are related to symmetries of the system. These may be Lie point symmetries but usually they are generalized symmetries and they form more general algebras than Lie ones. Our aim is to study and use Lie symmetries of difference equations and to discretize differential equations preserving their most important properties. These include their Lie point symmetries, generalized symmetries, integrability and superintegrability.

In order to do so we plan to host a top-class researcher from a Canadian first class laboratory, Prof. Pavel Winternitz of the Centre de recherches mathématiques, Université de Montréal, Monreal (PQ) Canada, an expert in the field of symmetry preserving discretization and construction of superintegrable systems. This will strengthen the host institution’s research skills and its relations with the laboratory of the researcher.

Coordination
Roma Tre University , VIA OSTIENSE 159
00146 ROMA, Italy

Researchers involved

Pavel Winternitz (University of Montreal) - Principal Investigator
Decio Levi (Roma Tre University ) - Host
Rutwig Campoamor Stursberg ( Universidad Complutense, Madrid)
Faruk Gungor ( Istanbul Technical University)
Antonella Marchesiello ( Czech Technical University, Prague)
Luigi Martina (Salento University )
Danilo Riglioni (Roma Tre University )
Miguel Angel Rodriguez ( Universidad Complutense, Madrid)
Masoumeh Sajedi (University of Montreal)
Libor Snobl ( Czech Technical University, Prague)

Research and dissemination activities carried out in the framework of this project

1. P. Winternitz partecipation to Conferences

i. Symmetry and Perturbation Theory (SPT2014), Cala Gonone, Nuoro, Italia, from May 25 till June 1, 2014. Poster, Program, Slides, Photo.


ii. International Conference on Integrable Systems and Quantum symmetries (ISQS), Prague, Czech Republic, from June 23 till June 29, 2014 Poster, Schedule, Slides.


iii. Exact Solvability and Symmetry Avatars. Conference held on the occasion of Luc Vinet's 60th birthday. Montreal, Canada, from August 25 till August 29, 2014, Poster, Schedule


iv. Analytical Mechanics & Differential Geometry in honour of the 70th birthday of Sergio Benenti, Torino, Italy, from March 11 till March 14, 2015.2015. Program, Slides.


v. XXth Edition of the NEEDS (Nonlinear Evolution Equations and Dynamical Systems) conference series Santa Margherita di Pula, Cagliari, Italy, from May 24 till May 31, 2015. Poster, Program, Book of Abstracts, Slides.


vi. Integrable Systems and All That 2015 ...waiting for PMNP 2015, Lecce, Italy, June 15th, 2015. Program, Slides.


vii. PMNP, Physics and Mathematics of Nonlinear Phenomena, 2015, Gallipoli, Italy, from June 20 till June 27, 2015. Poster, Program, Book of Abstracs, Photo, Slides.

2. visits of collaborators

i. Faruk Gongur, retired professor at Istanbul Technical University Faculty of Sciences and Letters, Department of Mathematics
34469 Maslak, İstanbul, Turkey
Period: 5-9/11/2014

ii. Javier de Lucas Araujo, Assistant professor at University of Warsaw
Department of Mathematical Methods in Physics (KMMF)
ul. Pasteura 5, 02-093, Warsaw, POLAND
Period: 10-14/11/2014

iii. Cristina Sardon Munoz, PhD student at the University of Salamanca
Department of Fundamental Physics. Area of Theoretical Physics.
Plz. de la Merced, s/n 37008, Salamanca , España
Period: 1-10/12/2014

iv. Miguel Angel Rodriguez, professor at Universidad Complutense
Departamento de fisica Teorica II, Metodos Matematico de la Fisica
av. Complutense, Madrid, España
Period: 11-12/12/2014

v. Masoumeh Sajedi, PhD student at Montreal University
Department of Mathematics and Statistical Sciences
Montreal, PQ, Canada
Periods: 3-10/3/2015, 15-22/5/2015

vi. Antonella Marchesiello and Libor Snobl, Czech Technical University
Department of Physics, Faculty of Nuclear Sciences and Physical Engineering
Břehová 7,115 19 Prague 1, Czech Republic
Periods: 7-23/5/2015

3. visits to collaborators

i. Antonella Marchesiello and Libor Snobl, Czech Technical University
Department of Physics, Faculty of Nuclear Sciences and Physical Engineering
Břehová 7,115 19 Prague 1, Czech Republic
Periods: 19-30/6/2014, 02-07/09/2014, 27/01-04/02/2015, 17/04-01/05/2015, 01-04/07/2015

ii. M. Sajedi and I. Marquette, CRM, Montreal University
Department of Mathematics and Statistical Sciences
Montreal, PQ, Canada
Periods: 01/7-01/09/2014, 13/12/2014-07/01/2015

iii. Miguel Angel Rodriguez, Universidad Complutense
Departamento de fisica Teorica II, Metodos Matematico de la Fisica
av. Complutense, Madrid, España
Period: 21-28/03/2015

iv. Luigi Martina, Salento University
Dipartimento di Matematica e Fisica "Ennio de Giorgi",
via per Arnesano - 73100 Lecce
Period: 03-28/06/2015

4. P. Winternitz articles published, submitted or in preparation

i. Alexander Bihlo, Xavier Coiteux-Roy and Pavel Winternitz, The Korteweg-de Vries equation and its symmetry-preserving discretization, arXiv:1409.4340, 2015 J. Phys. A: Math. Theor. 48 055201 doi:10.1088/1751-8113/48/5/055201 Preprint, Article.


ii. Decio Levi, Luigi Martina and Pavel Winternitz, Lie-point symmetries of the discrete Liouville equation, arXiv:1407.4043, 2015 J. Phys. A: Math. Theor. 48 025204 doi:10.1088/1751-8113/48/2/025204 Preprint, Article.


iii. V Dorodnitsyn, E Kaptsov, R Kozlov and P Winternitz, The adjoint equation method for constructing first integrals of difference equations, 2015 J. Phys. A: Math. Theor. 48 055202 doi:10.1088/1751-8113/48/5/055202 Article.


iv. Decio Levi, Luigi Martina and Pavel Winternitz, Structure preserving discretizations of the Liouville equation and their numerical tests, arXiv:1504.01953 Submitted to SIGMA Preprint


v. Antonella Marchesiello, Libor Snobl and Pavel Winternitz, Three-dimensional superintegrable systems in a static electromagnetic field, arXiv:1507.04632, submitted to J. Phys. A Preprint


vi. Rutwig Campoamor-Stursberg, Miguel A. Rodríguez and Pavel Winternitz, Symmetry preserving discretization of ordinary differential equations. Large symmetry groups and higher order equations, arXiv:1507.06428 submitted to J. Phys. A. Preprint


vii. S.Post and P.Winternitz,General Nth order integrals of the motion, arXiv:1501.00471, submitted to J. Phys. A. Preprint


viii. I.Marquette, M.Sajedi and P.Winternitz, Fourth order superintegrable systems separating in Cartesian coordinates, in preparation.


ix. J.de Lucas, C Sardon and P. Winternitz, Lie Hamilton systems related to transitive primitive Lie algebras, in preparation.


x. D.Riglioni, M.Sajedi and P.Winternitz, Superintegrable systems involving two particles with spin s=1/2, in preparation.


xi. Decio Levi, Luigi Martina and Pavel Winternitz, Lie-point symmetries of the discrete 3-wave equation, in preparation.

Outreach activities carried out in this Project

i. Partecipations to the European night of the researchers, September 26, 2014,
Largo San L. Murialdo,1 European Corner

ii. Seminar: Superintegrability and exact solvability. Concepts and perspectives.
20-01-2015 - 14:30 Dept. Math. Phys., Roma Tre University,
AULA 311 (SEMINARI) Largo San L. Murialdo,1 Abstract, Slides.

iii. Seminar: Superintegrability and exact solvability. Concepts and perspectives.
03-03-2015 - 15:00 Dept. Math. Phys., Roma Tre University,
AULA C - Via Della Vasca Navale 84, Colloquio di Fisica, Slides.

iv. Seminar: General N-th Order Integrals of Motion in Classical and Quantum Mechanics
25-03-2015 - 14:30 Facultad de Ciencias Fisicas, UCM,
AULA:Seminario Depto. Física Teorica II, Planta 2a Seminario Fisica Teorica II, Slides.

v. Seminar: General Nth order integrals of motion 21-04-2015 - 14:30 Doppler Institute Seminars, Technical University Prague, Doppler Institute Seminars, Abstract, Slides.

vi. Seminar: General Nth order integrals of the motion in classical and quantum mechanics
4-05-2015 - 13:00 Dip. Mat. Inf., Perugia University,
Aula 3, "Antonella Fiacca", The Department Seminars, Slides.

vii. Seminar: General N-th Order Integrals of Motion in Classical and Quantum Mechanics
15-05-2015 - 14:30 Dept. Math. Appl., Naples University "Federico II", Seminar Announcement, Slides.

viii. PhD Course in Physics, Roma Tre University: CLASSICAL AND QUANTUM SUPERINTEGRABILITY WITH APPLICATIONS Course Program, Time Table.

ix. PhD Course in Physics, Roma Tre University: CLASSIFICATION AND IDENTIFICATION OF LIE ALGEBRAS Course Program, Time Table, Slides.

x. The 24th Student Conference "Winter School on Mathematical Physics", Janské Láznì, Czech Republic, from January 25 till January 31, 2015. Poster, Program, Book of Abstracts.


Discretizzazione di sistemi integrabili, superintegrabili e non integrabili preservando le simmetrie.

FP7-PEOPLE-2013-IIF - Marie Curie Action: "International Incoming Fellowships"
Numero di riferimento del Progetto: 624199
Finanziato da: FP7-PEOPLE
Periodo di permanenza dell'esperto straniero: dal 2014-05-05 al 2015-07-04


Riassunto del Progetto

Lo scopo di questo progetto è quello di sviluppare ed applicare strumenti matematici moderni per lo studio dei fenomeni quantistici e classici in un ambito discreto. La motivazione è, da un lato quella che al livello della fisica di base sembra che lo spazio - tempo è discreto a causa della esistenza della lunghezza di Planck ed il suo ruolo ad esempio nella gravità quantistica. D'altra anche nel mondo continuo molti importante fenomeni sono discreti, come ad esempio i fenomeni che si verificano in cristalli o in catene molecolari o atomiche. Così un'equazioni alle differenze finite può essere più fondamentale di un'equazione differenziale. Inoltre, le equazioni differenziali spesso possono essere risolte solo numericamente e percio' devono essere discretizzate, cioè approssimate da un sistema alle differenze finite.

Il nostro interesse principale è nei modelli che possono essere risolti esattamente a causa delle loro proprietà di simmetria e di integrabilità. Di particolare interesse sono i modelli integrabili e superintegrabili a dimensione finita ed infinita. I sistemi integrabili hanno tanti integrali primi del moto quanti sono i gradi di libertà (che possono anche essere infiniti). I sistemi superintegrabili hanno più integrali del moto dei gradi di libertà e questi integrali formano interessanti algebre non abeliane. Gli integrali del moto sono correlati con le simmetrie del sistema. Queste possono essere simmetrie puntuali di Lie, ma di solito sono simmetrie generalizzate e formano algebre più generali di quelle di Lie. Il nostro obiettivo è quello di studiare e utilizzare simmetrie di Lie di equazioni alle differenze per discretizzare le equazioni differenziali preservando le loro proprietà più importanti. Queste includono le loro simmetrie puntuali di Lie, simmetrie generalizzate, integrabilità e superintegrabilita'.

Per fare questo abbiamo chiamato un ricercatore di alto livello da un centro di ricerca d'eccellenza canadese, il Prof. Pavel Winternitz del Centre de recherches mathématiques, Université de Montréal, Montreal (PQ) Canada, esperto nel campo della discretizzazione preservando le simmetrie e nella costruzione di sistemi superintegrabili. Ciò rafforzerà le capacità di ricerca dell'istituto ospitante e le sue relazioni con il laboratorio del ricercatore ospitato.

Coordinamento
Universita' Roma Tre, VIA OSTIENSE 159
00146 ROMA, Italia