Nonlinear Conservation laws and applications

The theory of nonlinear conservation laws arises as a general mathematical framework for the description of a variety of mathematical structures in classical mehcanics, differential geometry, singularity and catastrophe theory and nonlinear phenomena in fluid dynamics, quantum physics, nonlinear optics, statistical mechanics and thermodynamics of classical fluids, liquid crystals, random graphs, complex and biological networks.

Main research directions are:

1. Classification of (multidimensional) nonlinear consevation laws realised via nonlinear partial differential equations (PDEs) of hydrodynamic type with viscosity, dispersion or dispersionless, universal behaviour of solutions and singularities.

For 1+1 dimensional systems, the approach extends the classical symmetry method combined with the Dubrovin's perturbative approach to the classification of Hamiltonian PDEs. Besides the classification results the method is also effective for the construction of solvable asymptotic equations for the decription of the critical behaviour of solutions in the vicitiy of gradient catastrophe points [collaborations: A. Arsie (Toledo, Ohio, USA), P. Lorenzoni (Milano-Bicocca, Italy); B. Dubrovin (SISSA, Italy), T. Grava (SISSA, Italy and Bristol, UK), C. Klein (Bourgogne, France); S. Trillo (Ferrara, Italy)].
For 2+1 dimensional systems the perturbative classification approach builds up on Ferapontov's definition of integrability via hydrodynamics reductions. The classification allows to construct integrable PDEs by the request that perturbed equation inherits all hydrodynamic reduction of their dispersionless limit [Collaborations: E. Ferapontov (Loughboroguh, UK), V. Novikov (Loughborough, UK) ].

2. Complete integrability and equations of state of mean field statistical mechanical models and random networks .

A wide class of mean field models, e.g. magnetic, fluid systems, liquid crystals and random networks can be solved via the construction of differential identities for the partition function of suitably deformed models. Equations of state are obtained as solutions of a nonlinear conservation law with a initial condition that specifies the models. The project aims a developing a general method/approach where the theory of nonlinear conservation laws allows for the construction of equations of state for multicomponent statistical mechanical models of interest in statistical physics, random networks and bio-chemistry [Collaborations: E. Agliari (Rome, Italy), A. Barra (Salento, Italy), C. Benassi (Northumbria), M. Dell'Atti (Northumbria), L. Dello Schiavo (Bonn, Germany), G. De Nittis (Santiago, Chile), F. Giglio (Newcastle, UK), G. Landolfi (Salento, Italy), P. Lorenzoni (Milano-Bicocca, Italy), O. Senkevich (Northumbria, UK)].

3. Integrability, Toda Lattice, KP and complexity in random matrix models. .

This research lies at the crossroad between the study of integrable systems, statistical mechanics and random matrix theory. Based on the study of wide class of matrix models and their asymptotic regimes, the aim it to describe emergente complexity through the analysis of phase diagrams associated to the critical behaviours of dispersive conservatio laws. The phenomenology is expected to be fundamentally different from the classical mean field statistical mechanical models. An essential ingredient of the proposed analysis is represented by the well established relationship between random matrix models and the theory of nonlinear integrable systems. combined with the methods of conservation laws that have been successfully applied to the fluid and magnetic models, liquid crystals models. In particular, the study of random matrix models involves nonlinear integrable lattice equations (e.g. Toda lattice) and multidimensional dispersive integrable equations (e.g. Kadomtsev-Petviashvili equation) [Collaborations: C. Benassi (Northumbria, UK), G. Biondini (Buffalo, NY, USA), B. Prinari (Colorado Springs, USA )].