Dottorato in FISICA

Titolo della tesi: Towards the theory of anomalous waves in nature and nonlinear Schrödinger type equations.

The focusing Nonlinear Schrödinger (NLS) equation i ψ_t+ ψ_xx+|ψ|^2 ψ =0 with ψ ( x , t ) ∈ C is the simplest integrable model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, and it is relevant in various areas of physics. Its elementary solution ψ_0 ( t ) =a exp( 2 i |a|^2 t) is unstable under the perturbation of waves with sufficiently large wave length, and this MI is considered as the main physical mechanism for the formation of anomalous waves (AWs) in nature. During the last few years the periodic Cauchy problem of NLS for generic initial perturbations of the background solution ψ_0 (the so-called Cauchy problem of AWs) has been solved by P. M. Santini e P. G. Grinevich, to leading and relevant order, in terms of elementary functions of the initial data (if the number of unstable nonlinear modes of the problem is sufficiently small), using the "finite gap method". The corresponding dynamics gives an analytic description of the recurrence of nonlinearly interacting unstable modes in terms of elementary functions of the initial data and, in the simplest case of one unstable mode only, one obtains the analytic and quantitative description of a Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence of AWs. This result opens several interesting research directions. The aim of this thesis is twofold: i) the analytic study of the AW dynamics in nonlinear lattices and in classical relativistic field theories; ii) the analytic study of the effect of perturbations, like a small dissipation, on the dynamics of periodic AWs. The thesis is organized in the following three parts. 1) in chapter 2 the perturbation theory allowing one to study the effects of perturbations on the FPUT recurrence of NLS AWs is constructed, and the theory is applied to two physically relevant generalizations of NLS: the complex Ginzburg-Landau and the Lugiato-Lefever equations. 2) In chapter 3 we study the AW dynamics in the Ablowitz-Ladik lattice, an integrable discretization of the NLS equation. We describe the FPUT recurrence of AWs and we apply the perturbation theory developed in chapter 1 to the study of the effect of a small loss or gain in the dynamics of AWs on the lattice, emphasizing similarities and differencies with respect to the continuous case. 3) In chapter 4 we study the generalization of NLS to a classical relativistic field theory: the Massive Thirring Model (MTM), an integrable relativistic field theory reducing to the Klein Gordon theory in the linear limit, and to the focusing and defocusing NLS equations in the non relativistic limit. We construct, using Darboux-Backlund transformations, a large phenomenology of exact solutions and we study their energy (i.e., if they correspond to particles or anti-particles) and velocity (i.e., if they propagate inside or outside the light cone (tachyons)). Moreover we study two examples of FPUT recurrences of AWs of the model, emphasizing similarities and differencies with respect to the non relativistic case.