## Final Report Summary - SPEDIS (Symmetry preserving discretization of integrable, superintegrable and nonintegrable systems)

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

The motivation was 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.

The main interest is in models that can have symmetry and integrability properties, in particular 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. The 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.

The visit of P. Winternitz gave rise to a set of interactions between research groups situated in various regions of the European Community characterized by the fact of collaborating with him on the various aspects of this project. During his stay there have been visits in Roma Tre of researchers from Warsaw, Prague and Montreal interested in superintegrable models, from Madrid and Lecce on symmetry preserving discretization. The work interactions of Winternitz were not limited to the visits but were carried out also by Skype with researchers in Canada, Russia and USA and by his visits to Canada, Lecce, Madrid and Prague.

These interactions, occurring during the visit of P.Winternitz to Rome, gave rise to 11 scientific manuscripts (3 published in journals, 4 submitted to journals and published in the Preprint Archive http://arxiv.org and 4 in preparation). Publications (1,2,4) and (11) are devoted to a new field, namely the symmetry preserving discretization of partial differential equations and have practical applications in the field of geometrical integration. The articles (2,4) and (11) concern equations with infinite dimensional symmetry algebras. Manuscripts (3,6) and (9) concern nonlinear ordinary differential and difference equations: the construction of first integrals (3), discretization (6) and nonlinear superposition formulas (9). Finally manuscripts (5,7,8) and (10) are devoted to new types of superintegrable systems. They involve particles in magnetic fields (5), particles with spin (10), or higher order integrals of motion (7,8).

All these new results are interesting in themselves and are the starting point for new activities in this field in the research groups in Roma Tre University, in the Universidad Complutense, in the Czech Technical University, in Warsaw University and in the University of Salento. This is evidenced by the presence of 4 works in preparation (8-11).

The PhD School in Mathematics and in Physics took advantage of P. Winternitz’s presence in Rome to increase the cultural offer to its PhD students with two courses on “Classical and Quantum Superintegrability with Applications” and on “Classification and Identification of Lie Algebras”. Both subjects of great interest for theoretical and mathematical physicists.

The main impact of this visit is by certainly scientific. It gave rise to scientific results which will remain to indicate the commitment of the EC in the progress of pure science. This was very important for the diffusion of scientific ideas. P.Winternitz during his stay, gave 13 seminars at different Universities in Europe and during International Conferences or Workshops in the field. A long range impact will be due to his PhD courses which transferred information previously not available in Rome scientific community.

More detailed information on the results of this project are included in the web page: http://webusers.fis.uniroma3.it/~levi/SPeDISweb/.

References:

1. A. Bihlo, X. Coiteux-Roy and P. Winternitz ( 2015) The Korteweg-de Vries equation and its symmetry-preserving discretization, arXiv:1409.4340 J. Phys. A: Math. Theor. 48 055201 doi:10.1088/1751-8113/48/5/055201

2. D. Levi, L. Martina and P. Winternitz ( 2015) Lie-point symmetries of the discrete Liouville equation, arXiv:1407.4043 J. Phys. A: Math. Theor. 48 025204 doi:10.1088/1751-8113/48/2/025204

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

4. D. Levi, L. Martina and P. Winternitz, Structure preserving discretizations of the Liouville equation and their numerical tests, arXiv:1504.01953 Submitted to SIGMA. (Symmetry, Integrability and Geometry: Methods and Applications).

5. A. Marchesiello, L. Snobl and P. Winternitz, Three-dimensional superintegrable systems in a static electromagnetic field, arXiv:1507.04632 submitted to J. Phys. A: Math. Theor.

6. R. Campoamor-Stursberg, M. A. Rodríguez and P. Winternitz, Symmetry preserving discretization of ordinary differential equations. Large symmetry groups and higher order equations, arXiv:1507.06428 submitted to J. Phys. A: Math. Theor.

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

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

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

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

11. D. Levi, L. Martina and P. Winternitz, Lie-point symmetries discretization of the 3-wave equation, in preparation.

The motivation was 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.

The main interest is in models that can have symmetry and integrability properties, in particular 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. The 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.

The visit of P. Winternitz gave rise to a set of interactions between research groups situated in various regions of the European Community characterized by the fact of collaborating with him on the various aspects of this project. During his stay there have been visits in Roma Tre of researchers from Warsaw, Prague and Montreal interested in superintegrable models, from Madrid and Lecce on symmetry preserving discretization. The work interactions of Winternitz were not limited to the visits but were carried out also by Skype with researchers in Canada, Russia and USA and by his visits to Canada, Lecce, Madrid and Prague.

These interactions, occurring during the visit of P.Winternitz to Rome, gave rise to 11 scientific manuscripts (3 published in journals, 4 submitted to journals and published in the Preprint Archive http://arxiv.org and 4 in preparation). Publications (1,2,4) and (11) are devoted to a new field, namely the symmetry preserving discretization of partial differential equations and have practical applications in the field of geometrical integration. The articles (2,4) and (11) concern equations with infinite dimensional symmetry algebras. Manuscripts (3,6) and (9) concern nonlinear ordinary differential and difference equations: the construction of first integrals (3), discretization (6) and nonlinear superposition formulas (9). Finally manuscripts (5,7,8) and (10) are devoted to new types of superintegrable systems. They involve particles in magnetic fields (5), particles with spin (10), or higher order integrals of motion (7,8).

All these new results are interesting in themselves and are the starting point for new activities in this field in the research groups in Roma Tre University, in the Universidad Complutense, in the Czech Technical University, in Warsaw University and in the University of Salento. This is evidenced by the presence of 4 works in preparation (8-11).

The PhD School in Mathematics and in Physics took advantage of P. Winternitz’s presence in Rome to increase the cultural offer to its PhD students with two courses on “Classical and Quantum Superintegrability with Applications” and on “Classification and Identification of Lie Algebras”. Both subjects of great interest for theoretical and mathematical physicists.

The main impact of this visit is by certainly scientific. It gave rise to scientific results which will remain to indicate the commitment of the EC in the progress of pure science. This was very important for the diffusion of scientific ideas. P.Winternitz during his stay, gave 13 seminars at different Universities in Europe and during International Conferences or Workshops in the field. A long range impact will be due to his PhD courses which transferred information previously not available in Rome scientific community.

More detailed information on the results of this project are included in the web page: http://webusers.fis.uniroma3.it/~levi/SPeDISweb/.

References:

1. A. Bihlo, X. Coiteux-Roy and P. Winternitz ( 2015) The Korteweg-de Vries equation and its symmetry-preserving discretization, arXiv:1409.4340 J. Phys. A: Math. Theor. 48 055201 doi:10.1088/1751-8113/48/5/055201

2. D. Levi, L. Martina and P. Winternitz ( 2015) Lie-point symmetries of the discrete Liouville equation, arXiv:1407.4043 J. Phys. A: Math. Theor. 48 025204 doi:10.1088/1751-8113/48/2/025204

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

4. D. Levi, L. Martina and P. Winternitz, Structure preserving discretizations of the Liouville equation and their numerical tests, arXiv:1504.01953 Submitted to SIGMA. (Symmetry, Integrability and Geometry: Methods and Applications).

5. A. Marchesiello, L. Snobl and P. Winternitz, Three-dimensional superintegrable systems in a static electromagnetic field, arXiv:1507.04632 submitted to J. Phys. A: Math. Theor.

6. R. Campoamor-Stursberg, M. A. Rodríguez and P. Winternitz, Symmetry preserving discretization of ordinary differential equations. Large symmetry groups and higher order equations, arXiv:1507.06428 submitted to J. Phys. A: Math. Theor.

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

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

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

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

11. D. Levi, L. Martina and P. Winternitz, Lie-point symmetries discretization of the 3-wave equation, in preparation.