### **General description of the project**

During the first 6 months of 2022 there was an intense activity related to the expansion of the Iniziativa Specifica MMNLP and the corresponding broadening of its activities. In particular, the following steps have been undertaken:

- The new Unit of Trieste has been added to MMNLP. The members who are also permanent staff members are T. Grava (local coordinator, SISSA), M. Bertola (SISSA), D. Guzzetti (SISSA). There are also 3 PhD students. The association of a further member, D. Lewanski (U. Trieste), is under way. This ensures the presence of MMNLP in the Sezione INFN of Trieste, which is quite an important centre for Theoretical Physics in Italy. We invite the Readers tohave a look at the section with the report of activities in 2022 and the project of activities in 2023 in order to see how wide are the research interests of the new Unit and how they are deeply embedded in mainstream topics in Theoretical Physics.
- New permament staff members have been added to the Unit of Rome: A. De Sole and D. Valeri, together with one PhD student. This adds vitality and broadens the research topics of the Unit, in view of continuing the MMNLP project for the next three-years period 2023-2026. The newly added members are long-standing collaborators of V. Kac and, together with him, they developed an original algebraic approach to Integrable Systems which is bringing new insights and results to the field. The association of a further member, M. Falessi (ENEA - permanent staff member), is under way.

The aim of the expansion is to consolidate the MMNLP project and to make it the **reference project in the field of Integrable Systems in Italy**. At the moment, to our knowledge, MMNLP gathers almost all Italian high-level researchers whose interests are in this field.

**Scientific activities of the various Research Units**

### Lecce Unit

**1. Geometry of non local Poisson brackets and integrable systems (Vitolo). **

Weakly non-local Hamiltonian operators of hydrodynamic type play an important role in the theory of integrable PDEs. Despite their rich geometry and their ubiquitous pre-sence their use so far has been quite limited, probably due to high computational difficulties. Remarkably, recent results (collaboration with M. Casati, P. Lorenzoni, D. Valeri) allow to considerably reduce such difficulties and open the possibility to study further geometric properties of such operators and of the associated (bi)-Hamiltonian hierarchies (new examples found in collaboration with E. Ferapontov, M. Pavlov and PhD students J. Vasicek, P. Vergallo).

An important goal (in collaboration with P. Lorenzoni and S.Shadrin) is to compute the Hamiltonian and bi-Hamiltonian cohomologies in analogy to what has been achieved in previous works in the local case. In order to complete this task a four-months visit of R. Vitolo to S. Shadrin at the U. van Amsterdam is planned.

A formulation of non-local operators as local operators on jet spaces with constraints is another goal aimed at a deeper mathematical understanding of such Poisson brackets (collaboration with Krasil’shchik and Verbovetsky).

**2. Integrable models in low energy effective field theories (G. De Matteis, G. Landolfi, L. Martina).**

In many low energy effective field theories the competition of long/short range interactions lead to collective localized excitations, generically described by non integrable PDEs. But all modern mathematical tools from the integrable systems and topological arguments can be applied in order to understand their basic structures, stability and dynamical behavior.

Thus, we will consider generalized nonlinear sigma-models, extended Skyrme models, multi-component/dimensional Landau-Ginzburg equations, generalized Burger’s equations, Einstein equations in cosmic string background, from point/generalized and asymptotic symmetries properties, hamiltonian structures, Haantjes operators, multi-scale approximation, Padé approximants of Painlevé equations (with F. Zullo, Roma unit), perturbative regularization techniques.

The objectives: a) characterize the integrable sectors of the above models, b) study the interplay among integrable sector solutions and localized states of the full theory, c) describe, in perturbative sense, the interactions of localized excitations among themselves and with space extended solutions (like domain walls and magnons). Applications to several physical context, ranging from low energy effective QCD to condensed matter and stellar structures are envisaged. (Collaboration with C. Naya, V. Turco, A. Moro, F. Giglio, F. Capone.)

**3. Integrable systems, tropical geometry and applications (Angelelli).**

Research activities will focus on the study of combinatorial and information-theoretic properties coming from assumptions on a special class of algebraic expressions. These models have direct applications in the study of properties of Wronskian solutions of some integrable systems (e.g., KP II equation) and, specifically, in their "tropical limit".

Special attention will be paid to the complexity associated with algebraic expansions of determinantal solutions (e.g. Cauchy-Binet expansion) under given algebraic assumptions. These concepts will be explored also in the framework of contextuality and tropical structures in statistical physics (micro-macro correspondence, matrix ensembles, ensemble non-equivalence, systems out of the equilibrium). (Collaboration with B. Konopelchenko.)

### Rome Unit

**1. Towards an analytic theory of anomalous (rogue) waves in nature (Santini).**

Anomalous waves (AWs) are anomalously large waves with respect to the average ones; they appear and desappear in an apparently unpredictable way, and they can be very dangerous in oceanography. They are ubiquitus in nature and their physical origin is the nonlinear modulation instability of some basic background solutions, like the Stokes waves in the water wave theory.

The simplest nonlinear model describing such phenomena to leading order is the focusing NLS equation. i) The Cauchy problem for periodic generic perturbations of the unstable background solution of NLS has been recently solved with Grinevich, adapting in a significant way the Finite Gap method (a nonlinear analogue of the Fourier method) to the AW problem, in the case of a finite number of nonlinear unstable modes. ii) The main effects of a small loss or gain on the AW dynamics have been analytically described (with Grinevich and Coppini).

These results open several directions of research, that we plan to investigate in the project. In particular: a) the study the AWs in the case of a large number of unstable modes, a typical situation in the ocean, combining the deterministic aspects of the above theory with the tools of statistical mechanics (in collaboration with Grinevich). b) AWs and their dynamics in other contexts: on lattices and in relativistic field theories (in collaboration with Coppini). The envisaged goal is to construct a comprehensive analytic theory of Aws.

**2. Non commutative integrable evolution equations in Banach spaces and Bäcklund transformations (Carillo).**

The relevance of Bäcklund transformations (BTs) in studying nonlinear evolution equations is well known both under the viewpoint of finding exact solutions as well as in giving insight in the study of their symmetries, conserved quantities and Hamiltonian structure. The subject of the proposed study are non commutative evolution equations, in which the unknown is an operator on a Banach space, and their BTs.

The aim of the present research project is twofold: a) Reveal new structural properties of such equations, e.g. symmetry properties; b) Investigate possible Hamiltonian and bi-Hamil-tonian structures in the non-Abelian case (collaboration with Falqui, Pedroni, and Santini); c) Construct solutions of such equations. In particular, in the special case of finite dimensional operators, matrix solutions can be explicitly constructed. These particular cases turn out to be relevant in applications (collaboration with M. Lo Schiavo and C. Schiebold, F. Zullo). Among the applications a special mention is deserved by quantum mechanics (involving Colleagues of other Units).

**3. Bäcklund transformations and nonlinear PDEs; distribution of zeroes and poles of Airy functions and Painleve’ transcendents (Zullo).**

Two topics: a) Bäcklund transformations (BTs) play an important role in the construction of exact solutions of nonlinear PDEs; they represent exact discretizations of classical differential equations and, from the quantum point of view, can be seen as similarity transformations induced by the Baxter Q operator (equivalent of the classical canonical transformations) . b) For ODEs possessing entire (meromorphic) solutions with movable zeros (poles) there is the possibility to fix a solution by choosing the location of a zero (pole). Starting from it, it has been shown that i) solutions of the Painlevé equation I possess many properties very similar to those of the Weierstrass elliptic functions; ii) many characteristics of the trigonometric functions can be extended, in a natural way, to the solutions of the Airy equation (Airy functions and Painleve’ transcendents play an important role in many branches of modern mathematical physics).

Objectives : a) Use of BTs to construct explicit solutions of the underlying models, characterize their relevant properties and investigate their applications (e.g. to phenomena described by the Gross–Pitaevskii equation, the Emden-Fowler equation, the Ermakov-Pinney equation and the generalized Ermakov equations). Among the envisaged achievements of the proposed program is a detailed study of the analytic properties of the q-integral representation of the Baxter operator. This research on BTs partially involves another member of the unit: S. Carillo. b) Analytic description of the distributions of the zeros of Airy functions and of the poles of Painleve’ transcendents, through the use quasi-periodic properties. (collaboration with O. Ragnisco and H. Hone).

**Milano-Bicocca Unit**

**1. Integrable systems of topological type and weakly non local Poisson bracket (Lorenzoni). **

In the last 20 years a program of classification of 1+1 multi-component Hamiltonian system of conservation laws has been proposed by Dubrovin and Zhang. It is based on the notion of formal integrability and it is implemented by a perturbative scheme consisting in reconstructing the dispersive corrections to a quasilinear system of Hamiltonian conservation laws. Many important examples can be obtained starting from solutions of WDVV equations and the associated cohomological field theories. For instance, the polynomial solutions associated with Coxeter groups give rise to special integrable hierachies including the Drinfeld-Sokolov case. In the recent years the Dubrovin-Zhang perturbative approach has been extended to integrable quasilinear systems of conservation laws which do not admit (in general) any local hamiltonian structure.

The aims of this activity are: 1) To construct integrable hierarchies from solutions of generalized WDVV equations and the associated F-cohomological field theories (in collaboration with A. Arsie, A. Buryak and P. Rossi) Special cases we plan to study are those associated with polynomial solutions associated with complex reflection groups.The existence of these integrable systems relies on recent results that extend Givental's approach to generalized WDVV equations. 2) To study weakly non local Poisson bracket (WNLPB) and their deformations (in collaboration with S.Shadrin and R. Vitolo, see the description of the Lecce unit activities) and to develop symbolic computation packages for WNLPB (collaboration with M. Casati, D. Valeri and R. Vitolo).

**2. Stratified fluids and integrable and near-to-integrable models (Falqui, Ortenzi, Pedroni). **

One of the goals of this activity is to develop and analyse models of stratified fluid motion, and develop the study of suitable nonlinear PDEs such as the 1+1 dimensional model obtained by R. Camassa and W. Choi and more recently analysed in a series of papers. A significant part of this research task will be to identify the most important nonlinear interactions, to find simplified models, to study their Hamiltonian formulation(s) and to frame our results within the Dubrovin-Zhang perturbative expansion in the dispersion parameter.

Three cases will be considered: two-layer, multi-layer, and continuous stratification. It is expected that a detailed study of Hamiltonian systems on manifolds with boundaries will play an important role in this issue, as well as a deeper analysis of the available Hamiltonian description of the Euler equations via Clebsch variables. The extension of our results concerning the Hamiltonian structure of the non-Boussinesq dispersionless n-layer model to the (possibly weakly) dispersive case. The aim is to formulate suitable multiple scaling limit yielding (near-to-)integrable 1+1 dimensional averaged models also when dispersion is taken into account (collaboration with R.Camassa).

In oceanography, the classical hydrostatic approximation leads to modelize the internal waves with two-layer models that share the dispersionless part with NLS-like models. In collaboration with Rome Unit we plan to study wave motion in this context.

In collaboration with B. Konopelchenko we shall study and classify the classical shocks formation generally present in such systems of PDEs, as well as the instabilities generated by hyperbolic-to-elliptic transitions.

**3. Linear ODEs and quantum integrable systems (Raimondo).**

The ODE/IM correspondence turns out to be related with many aspects of modern mathematical physics, such as supersymmetric Gauge theories, affine Toda field theories, non-linear sigma models, generalized KdV theories, Hitchin integrable systems, and the geometric Langlands correspondence. The aim of the project is to complete the proof, started in recent years, of the following conjecture by Feigin and Frenkel. Let g be an affine Kac-Moody algebra. To any state of the quantum g-KdV model there exists a unique differential operator (more precisely, an oper), with values in the Langlands dual Lie algebra of g, whose generalized monodromy data satisfy the Bethe Ansatz equations for the given state of the quantum model (collaboration with D. Masoero and D. Valeri) .

**Trieste Unit**

The research activity of the Unit of Trieste uses methods of integrable systems in applications to a wide range of topics, including

- Symplectic geometry of moduli spaces and character varieties
- Inverse problems and spectral theory
- Integrable systems in statistical Mechanics and Random Matrices
- (Multi)-Orthogonal polynomials and functions
- Nonlinear waves
- Isomonodromic deformations and Frobenius manifold
- Cohomology of moduli spaces of curves, Gromov-Witten theory, integrable hierarchies and topological recursion.

Students:

- Luis Felipe Lopez Reyes (Ph.D. expected 09/2022)
- Eduardo Chavez Heredia (Ph.D. expected 09/2022)
- Massimo Gisonni (Ph.D. expected 09/2022)
- Giuseppe Orsatti (Ph.D. expected 09/2023)
- Dmitry Rachenkov (Ph.D. expected 09/2025)
- Master Student: Christian Puntini