Dipartimento di Matematica "Giuseppe Peano"

This project focuses on nontrivial solutions of equations and systems of differential equations characterized by strongly nonlinear interactions. We are interested in the effect of the nonlinearities on the emergence of non trivial self-organized structures. Such patterns correspond to selected solutions of the differential problem possessing some special symmetries or shadowing particular shapes. We wish to investigate the main analytical mechanisms involved in this process in terms of the global structure of the problem, exploiting their common perturbative-variational structure. By way of example, we intend to address the following themes:

 

A. Reaction diffusion systems, where pattern formation is driven by strong interactions. Our goal is to capture the geometry of the phase segregation and vortex formation. We deal with systems of differential equations with strongly competing interaction terms, modeling both the dynamics of competing populations (Lotka-Volterra) systems and other relevant physical phenomena, among which the phase segregation of solitary waves of Gross-Pitaevskii systems arising in the study of multicomponent Bose-Einstein condensates and Liouville type systems of interest in non abelian Chern-Simons vortices.

 

B. Existence and qualitative properties of solutions to nonlinear equations arising in several physical models, such as quantum field theory, cosmology and molecular physics, including vortices in the Chern-Simons and Ginzburg Landau models, vortex sheets, droplets and other patterns, also in the presence of nonlocal diffusions, nonlinearities, point defects.

 

C. Concentration phenomena for elliptic equations and systems arising in geometry and  physics. A typical situation occurs when, for some limiting values of a parameter, special solutions exhibiting a singular limiting behavior appear. This feature naturally occurs in many models, such as the nonlinear Schrödinger equation in the semi-classical limit, systems modeling biological pattern formation such as the gradient theory of phase transitions by Allen-Cahn and Cahn-Hilliard, and also in conformal geometry, where singular behavior arises in the form of bubbling triggered by the critical nature of the nonlinearities.

 

D. The N-body problem, where we focus on the construction of complex dynamics by juxtaposing (quasi)periodic solutions and parabolic arcs, as well as on their stability properties. Our final aim is to exploit alternatively KAM theory and variational methods to detect either regularity of motions or the occurrence of chaos.

 

E. Analysis of the evolution of the principal structures in infinite dimensional phase space for Hamiltonian and dissipative PDE's, including periodic orbits, embedded invariant tori and other structures, center manifolds, as well as stable and unstable manifolds, with further applications to pattern formation in nonlinear parabolic equations, the Schrödinger, Euler and the wave equations.