# Project Overview

**Background**

Partial Differential Equations (PDEs) undoubtedly are among the main tools for an efficient modelling of physical phenomena. Integrable PDEs, exhibiting infinite-dimensional analogues of regular (integrable) behaviour displayed by finite dimensional systems, began to be studied in the middle of the XX century in fluid dynamics, field theory and plasma physics.

It was observed that, in specific models and regimes, a balance between non-linearity and dispersion leads to the formation of stable patterns. The modern theory of integrable systems grew up around the study of the Korteweg de Vries (KdV) equation, with origins in the seminal work of Zabusky and Kruskal about the recurrence behaviour of solutions, the discovery of the Lax pair formulation, multi-soliton so-lutions and infinite number of conservation laws.

Over the last 30 years refined analytical and geometrical tools have been developed to study integrable or nearly integrable systems such as Riemann – Hilbert boundary values problems, infinite-dimensional generalizations of KAM and Nekhoroshev theories, algebraic geometry of Riemann surfaces and theta-functions, the theory of infinite-dimensional Lie algebras and techniques of differential geometry involved in the study of the Hamiltonian structures of dynamical systems and in the theory of Frobenius manifolds. The idea that an integrable behaviour persists in non-integrable systems, together with the combination of the state-of-the-art numerical methods with front-line geometrical and analytical techniques in the theory of Hamiltonian PDEs, is the leitmotiv of this research project. The asymptotic regimes leading to phase transitions display universality properties which can be analysed both numerically and analytically. The predictive power of numerics and scientific computing will be used both as a testing tool for theoretical models and as a generator of new conjectures.

Random matrices, introduced in the study the statistical properties of quantum systems with a large number of degrees of freedom (heavy nuclei), have been shown to successfully enter various fields such as number theory, combinatorics, quantum field theory, and even wireless telecommunications. One of the main reasons for their fascinating modelling power is that as the dimensions of the matrices tend to infinity the local statistical properties of the eigenvalues become independent of the probability distribution of the given matrix space. This important and long conjectured property of such a fast developing field of research was proved only recently. Like the class of dispersive PDEs mentioned above, these random “objects” display in suitable asymptotic scalings, a completely integrable behaviour. This was the clue to the conjecture, originally due to B. Dubrovin, that critical behaviour of Hamiltonian dispersive PDEs (near a gradient catastrophe of their dispersionless counterpart) is structurally independent of its dispersive “tail” and of the initial data. The critical behaviour is indeed conjectured to be universal, in the same way as the local statistical properties of random matrices become independent of the probability distribution when the size of the matrix goes to infinity.

**Objectives**

The principal scientific aim of the Network is to create a fertile research environement for scientists (especially Ph.D. students and Post/docs) working in the fields of Mathematical Physics and Numerical Applications thereof by means of a steady and intense flow of personnel exchanges between the European and the Overseas nodes of the Network, as well as the organization of Network Workshops. It is expected that the broad interdisciplinary basis and intertwining of methods of Geometry and Mathematical Physics lead to solutions of hard geometrical problems and to formulations of new mathematical models of physical phenomena.

The research programme is structured along three scientific Work Plans.

- (Near-to) Integrable PDEs and applications (WP2)
- Isomonodromic deformations, PainlevĂ© equations and random objects (WP3)
- Numerical approaches (WP4)

The primary research objectives are the following:

- Further understanding of the Mathematics of Integrability by completing the classification of the systems of spatially one-dimensional integrable evolutionary Partial Differential Equations (PDEs) that admit hydrodynamic limit (WP2).
- Understanding of the analytical aspects of asymptotic integrability of systems depending on a parameter, especially in the realm of the theory of water waves equations arising from suitable reductions of Euler flows (WP2).
- Understanding of the general universality classes appearing in critical regimes of Matrix models as well as the computation of significant properties (WP3).
- Express the tau function associated with the general solution of a system of isomonodromic deformations via generalisations of the PainlevĂ© functions as a fredholm determinant (WP3).
- Implementation of geometric and symplectic integrators for integrable and almost integrable PDEs (WP4)
- Creation of an open source numerical library for PainlevĂ© equations (WP4).

**Approch and Methodology**

The overall methodological approach relies on combining analytical, geometrical and numerical techniques, so far often confined within different areas of Mathematics and Physics. The research problems will be at firrst addressed by the approach of pure mathematics, based on the careful analysis of intrinsic connections between the mathematical constructions under consideration, on the precise formulation of the arising conjectures and on proving them as rigorous theorems. The power of efficient numerical tools will be instrumental to dismiss wrong conjectures, to formulate new ones and even to support some steps for their analytical proof. General tools of the theory of integrable systems such as Riemann – Hilbert boundary value problems, bihamiltonian geometry, Frobenius manifolds, as well as analytical tools such as general complex analysis, potential theory in the complex plane, Deift-Zhou steepest descent method for oscillatory integrals will be employed in the pursue of our goals.

**Dissemination Strategy**

Along with the three research Work Packages briefly described above, our work programme includes dissemination and exploitation strategies targeted to different levels.

**Academia:**

The dissemination plan for the Network research results will be targeted to the widest possible research community and will involve different actions, including publications, communications at Conferences, Workshops as well at running Seminars and organization of the three Network Conferences. We are confident that our results will be published in high level peer reviewed international journals. All publications will be made freely accessible either via open access or via preprint servers. To reach a bigger academic public, we plan to edit topical issues of renown journals to publish refereed proceedings of the major Network Events. Of special interest is the development of a library of numerical programs which will at first tested by the members of the Network, but finally be freely accessible for the community.**Outreach:**

In the Network’s life-span we plan to organize several lectures aimed at High School students and teachers, at least on those topics which better lend themselves to this scope, such as wave motion, geometry, and the combinatorial basic aspects of Random Matrices. Presentations will possibly occur at the MEETmeTONIGHT, Math en Jeans, and La nuit des chercheurs Science feasts, and at the institutional “Open Days”. Visits to local high schools to offer students a direct contact with the world of science are envisaged. These presentations will include brief explanations, projections of movies from numerical simulations, and the realization of miniature scale demonstrative experiments of fluid motion, to be performed with table-top tank equipped with generators.

IPaDEGAN deals with questions in mathematics and applications to theoretical physics; no ethical issues are involved in this project.