## Scientific activities of the various Research Units

### __ Lecce __

1) (Konopelchenko) Algebraic and projective differential geometric
aspects of the multidimenisonal dispersionless integrable
hierarchies. Hydrodynamic-type integrable systems describing
deformations of critical points of solutions of the linear Darboux
system. The tropical limit in statistical physics and in the theory
of integrable equations (with Angelelli).

2) (Landolfi) Integrable generalizations of thermodynamical models,
shocks and phase transitions (with Martina). Symmetries and
Entanglement of Quantum Systems.

3) (Martina) Interactions skyrmions-domain walls-magnons in the
Skyrme-Faddeev model. Chiral fermions in Souriau approach. Symmetry
preserving procedures (with Levi)

4) (Vitolo) Third-order local/nonlocal Hamiltonian structures of
Dubrovin-Novikov type and Mokhov operators. Integrable hierarchies
generated by triples of operators. Software techniques for nonlocal
integrability (with Martina). Quantization and symmetries in Lie
groups.

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1)
a) The development of a general approach to the integrable multi-dimensional quasi- linear PDEs of hydrodynamic type has shown a deep interrelation with the classical singularity theory, and it was proved that such PDEs determine the dynamics of critical points for certain classes of functions, related to non simple singularities from Arnold's list. Objectives are: to extend the previous results for two independent variables to the cases of arbitrary number of independent variables and also to integrable non- diagonalizable systems. In particular we will characterize the multi-dimensional extensions of the Gauss hypergeometric functions (Gelfand-Kapranov-Zelevinski equations) and the Lauricella type functions.
b) Hydrodynamic type systems can be of mixed hyperbolic-elliptic type: the studies of the behaviour of such two- and multi-component systems near the hyperbolic-elliptic transition line are in their infancy. The properties of the corresponding parabolic systems and their connection with integrable hydrodynamic type systems of Jordan form can be studied. The importance of particular multi-scaling expansions can be demonstrated. The problem of the Hamiltonian structures of mixed systems in the neighbourhood of a transition line can be addressed too. Objectives are: the development of different methods for the regularizations of the gradient catastrophes for such integrable hydrodynamic type systems, the understanding of the possible regimes of formation and development of instabilities in hydrodynamic type systems, in particular when non-diagonalizable systems of Jordan form appear, and the study of their algebraic and Hamiltonian properties. This study will allow to enlarge the class of integrable Hamiltonian hydrodynamic type systems, and finding connections with the, so-called, confluence process for multidimensional Lauricella type functions and other multidimensional Gauss hypergeometric type functions.
c) The description of N-dimensional integrable systems as the
closeness conditions for differential N-1 forms in the Grassmannians
Gr(N-1,N+1) provides compact forms to the corresponding hierarchies of
equations, like Liouville, dispersionless Kadomtsev- Petviashvili,
dispersionless Toda, Plebanski second heavenly equation and
others. Projectively dual Grassmannians define self-dual quasi-linear
systems. Objectives are: look for a unifying algebro-geometric setting
of the corresponding multidimensional dispersionless integrable
hierarchies of PDEs.

2)
a) In the last decades systems endowed with PT symmetry attracted a great interest for their applications in open quantum systems, although the method needs an adaptation of the standard QM framework. The investigation of the decoherence phenomena is one of the main arena of the PT-symmetric systems. Objectives are: the investigation of the integrability aspects of the PT-symmetric models and their possible value in the description of quantum decoherence phenomena.
b) Inspired by the recent developments in the study of the
thermodynamics of van der Waals fluids via the theory of nonlinear
conservation laws and the description of phase transitions in terms of
classical (dissipative) shock waves, we propose a novel approach to
the construction of multi-parameter generalisations of the van der
Waals model. Exploiting the theory of integrable nonlinear
conservation, we study a four parameter family of integrable extended
van der Waals models, linearisable by a Cole-Hopf
transformation. This family is further specified by the request that,
in regime of high temperature, far from the critical region, the
extended model reproduces asymptotically the standard van der Waals
equation of state. The extended van der Waals equation of state is
compatible with the notable empirical models, such as Peng-Robinson
and Soaveâ€™s modification of the Redlich-Kwong equations of state,
thus our approach also suggests that further generalisations can be
obtained by including the class of dispersive and viscous-dispersive
non-linear conservation laws and could lead to a new type of
thermodynamic phase transitions associated to nonclassical and
dispersive shock waves. Integrability properties of the van der Waals
equation will be investigated.

3)
a) Low energy excitations of the pure Yang-Mills theory, analog to a multi-component order parameter Landau-Ginzburg models, admit formulations in terms of certain nonlinear sigma-models (the Skyrme-Faddeev model, for instance) generically non integrable. These models admit finite energy knotted vortex-like structures (skyrmions), but also reductions leading to exact extended wave like solutions (magnons). Similar situations can be found in nemato-acoustic propagation, interpreted as a generalized version of the Skyrme Faddeev model. Objectives are: a) characterize the integrable sectors of the above models,
b) study the interplay among integrable sector solutions and generic initial data, in particular aimed to describe, perturbatively, the interactions of localized skyrmions with the extended â€œmagnonâ€ solutions.
c) Large classes of dynamical systems, not possessing local second degree Lagrangians, have been characterized by Souriau's geometrical approach. Hamiltonian formulation by non canonical Poisson brackets and self-consistent gauge fields both in configuration and momentum spaces were found. Correspondence with non relativistic anyons in 2D was obtained and 3D models were derived by semiclassical dynamics in crystals. Motion of electric charges in momentum monopole field was solved. Recently, also systems of chiral fermions, i.e. massless relativistic spin 1â„2 particles, have attracted a great interest both from physical and mathematical sides. The underlying semiclassical model has been deduced from the Weyl equation, simplifying the complicated quantum field theoretical calculations. Objectives are: to characterize the integrability conditions for such a class of models in/without the presence of external fields.

4) It was already observed that every third order Hamiltonian operator of Dubrovin- Novikov type in 1,2,3 dependent variables is the Hamiltonian operator of a hydrodynamic-type system. Some of the systems were already considered in the literature, and shown to be bi-Hamiltonian. There are evidences that such a result should hold in an arbitrary number of components. Furthermore, it is well known that a flat pencil of metrics generates a Frobenius manifold and hence an integrable hierarchy. Such pencils can be specialized using an additional Hamiltonian operator of the third order, following the old ideas by Olver and Rosenau. Objectives are: finding a complete classification of those operators, to prove that all of them are bi-Hamiltonian, and to understand the geometric nature of those systems with local and non-local Hamiltonian structures; finally we expect to obtain new integrable hierarchies, especially in the 3-component case.

### __ Roma I __

1) (Bruschi) Construction and investigation of i) classical/quantum
many- body models, sometimes isochronous, solvable by their
relationships with time dependent monic polynomials, and ii) of
integrable nonlinear matrix functional equations.

2) (Carillo) Noncommutative integrable matrix models by functional
analytical tools and BÃ¤cklund transformations.

3) (Santini) Rigorous aspects of the IST for nonlinear PDEs of
hydrodynamic type, and the dissipative and dispersive
regularizations of such PDEs. Recursion operators and BÃ¤cklund
transformations for hydrodynamic type models (with
Vitolo). Investigation of the role of modulation instabilities in
the formation of rogue waves for the focusing NLS model, studying
the evolution of a constant background subjected to small periodic
perturbations via algebro-geometric tools.

4) (Levi) Integrability/linearizability conditions for nonlinear
partial difference equations on the square lattice, or with more
than four lattice points per cell, via algebraic and/or multiple
scale analysis, and applications to dispersive
equations. Singularity confinement and exponential growth rate by
formal symmetry approach. Discretization of integrable immersion
formulae of surfaces in R3 and their applications (with
Martina). Symmetry preserving discretization of nonlinear partial
differential equations (with Martina). Construction of continuous and
discrete Darboux integrable systems and their linearization, Laplace
cascade, operator factorization (with Gubbiotti).

5) (Ragnisco) Many-body integrable and superintegrable
classical/quantum Hamiltonian systems on non-euclidean manifolds,
possibly with spin (part of the project with Latini). Integrable time
discretizations looking at the interpolating flows for the integrable
maps with n degrees of freedom associated with long-range spin chains,
and an analogous program in the quantum case. Discrete Quantum
Mechanics and the relation with orthogonal polynomials.

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1) Since the discovery and solution of the celebrated Calogero - Moser system, Calogero started a school devoted to the construction and investigation of classical and quantum nonlinear integrable systems and, in recent years, to the identification of isochronous systems (featuring in their phase space open regions where the solution evolves with fixed periods in all their degrees of freedom). Such systems have many applications (from chemistry to electrical circuits, to many-body physics, to cosmology, etc.), and their study also yields interesting mathematical findings (for instance, Diophantine relations satisfied by the zeros of named polynomials). Objectives are: to continue such a research i) investigating discrete analogues of the above systems and integrable many body problems with three body interactions; ii) exploiting a new technique based on the relations between the evolution of the zeroes and of the coefficients of monic, time-dependent polynomials, and iii) constructing and investigating novel integrable nonlinear matrix functional equations.

2 -3)
a) Integrable nonlinear PDEs of hydrodynamic type in multidimensions (like the dispersionless Kadomtsev-Petviashvili (dKP) and the heavenly equations) are the commutation condition of vector fields, and arise in various problems of Mathematical Physics and an adapted IST has been constructed, allowing one to solve their Cauchy problem. The associated nonlinear Riemann-Hilbert inverse problem can be effectively used to study the long time behaviour of solutions, exact implicit solutions, and the analytic aspects of wave breaking in multidimensions. Objectives are: the study of the rigorous aspects of such a IST, the application to other relevant models, like the dispersionless Davey-Stewartson (dDS) equation (integrable 2+1 dimensional analogue of the dispersionless nonlinear Schroedinger (dNLS) equation), and dissipative and dispersive regularizations (shocks) of multidimensional waves. Also multicomponent dNLS, the coupled Maxwell-Bloch system, and the integrable massive Thirring model can be analysed from such a perspective.
b) Some rational solutions of nonlinear integrable PDEs are considered
a good model of rogue waves, appearing in a large variety of physical
contexts. Results have been obtained on the algebraic construction,
via Darboux transformations, of rational solutions of a vector NLS and
of wave resonant interactions in media with quadratic nonlinearity as,
for instance, in anisotropic optical crystals. Objectives are: the
understanding of the physical and mathematical origins of rogue waves,
using the focusing NLS model, and studying the evolution of a constant
background subjected to small periodic perturbations.

4)
a) Difference equations play an important role in Physics and in Mathematics: on one hand they are believed to be engaged in the basic laws of physics (for example in quantum gravity formulation), on the other hand, they are the main ingredients of any numerical scheme. Thus, research areas are devoted to: a) the derivation of new integrable, or linearizable, nonlinear partial difference equations from discretized compatibility conditions and discretized classes of transformations, leaving them invariant; b) the development of symmetry invariant discretization procedures (Winternitz-Olver and Rebelo-Valiquette methods), with particular emphasis to nonlinear partial differential equations, in order to develop efficient discretization schemes for numerical purposes; c) the extension to discretized multidimensional geometrical objects, like surfaces and varieties, in terms of polytopes.
b) Integrable time-discretizations of finite dimensional classical and
quantum Hamiltonian systems deal with difference schemes preserving
the main features of continuous analogs such as, for instance, the
existence of closed orbits and possibly their stability under suitably
small perturbations. In this context, Suris unveiled the algebraic and
differential-geometric aspects of integrable discretizations, while
Kuznetsov and Sklianin investigated the role played by Backlund
transformations. Objectives are: integrable time discretizations
looking at the interpolating flows for the integrable maps with n
degrees of freedom associated with long-range spin chains, and then
performing an analogous program in the quantum case.

5) Integrable and superintegrable classical and quantum Hamiltonian systems constructed using co-algebra symmetries, though possibly not maximally superintegrable. Many new models of this type can be obtained by considering the intimate connection between interaction and metric. Remarkably enough, this superintegrability property also survives quantization, and Ä§-dependent quantum corrections only enter when the Schroedinger quantization is replaced by more sophisticated ones. Objectives are: the extension of these results to other classes of superintegrable systems.