Progetti di ricerca

The present proposal aims at studying nonlinear systems originating from classical and quantum physics that have the property of being exactly solvable or integrable in some asymptotic regime. For 50 years, integrability has continuously staked claims in an ever-growing range… Leggi tutto of applications in mathematics and in physics. Indeed, integrable and near-to-integrable models keep emerging in important areas of Classical and Quantum Physics such as Condensed Matter and Plasma Physics, Optics, Gravity, Statistical Physics and String Theories. 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 Kruskal and Zabusky about the recurrence behavior of solutions, the discovery of the Lax pair, multi-soliton solutions, bi-Hamiltonian structures and infinite number of conservation laws. In later surprising connections, integrable systems like the KdV equation and the Toda lattice were proven to appear in combinatorial models, in random matrices and the geometry of moduli spaces of algebraic curves. In general, integrability provides the route to an explicit description of behaviour and phenomena that turns out to be ubiquitous in nonlinear systems. Actually, the idea that an integrable behaviour persists in non-integrable systems, together with the combination of front-line geometrical and analytical techniques in the theory of Hamiltonian equations, is the leitmotiv of this research project, whose main aims can be summarized as follows: 1) To study geometric and algebraic properties of integrable Hamiltonian and bi-Hamiltonian systems arising in mathematical and physical models, together with their hierarchies of symmetries/conserved quantities; 2) To give new insights into the fundamental class of wave-induced dynamics in the theory of stratified fluids and in the theory of vorticity by linking their dimensionally reduced models to the theory of integrable PDEs; 3) To classify the universal emergent behaviours of nonlinear integrable PDEs in asymptotic regimes, by relating them to ”normal” forms, the most typical classes of which lead to nonlinear ODEs of Painlevé type; 4) To explore the connections between the integrability properties of Nonlinear Sigma models and the theory of Ricci flows. 5) To push further the theory of soliton gases, and further substantiate the features of the “unlikely marriage” of integrability and randomness motivated by the complexity of many nonlinear wave phenomena; 6) To use spectral methods and Riemann-Hilbert methods to describe solution for physically relevant initial/boundary value problems of nonlinear PDEs; 7) To study nonlinear wave phenomena, such as the formation and the dynamics of singularities, blow ups, wave breaking, shocks, elliptic/hyperbolic regime transition, dispersive/dissipative regularization.