TWO RESULTS ON THE GLOBAL DYNAMICS OF PDES ON THE CIRCLE.

This thesis is organized into two main parts, each addressing a distinct problem in the study of global in time approximation for two nonlinear Schrödinger equations. Chapter 1 focuses on the non-relativistic limit of the one-dimensional cubic nonlinear Klein-Gordon equation, which is known (formally) to converge to the cubic nonlinear Schrödinger equation as c → ∞. More precisely, it is known that smooth solutions of the Klein-Gordon equation converge over compact intervals of time to solutions of the nonlinear Schrödinger equation. This chapter is a joint work with D. Bambusi and F. Giuliani. Our main theorems show that, after an explicit Gauge transformation, one can construct a family of small-amplitude quasi-periodic solutions of the Klein-Gordon equation in H^s, the Sobolev space of s-regularity, with s large enough. This family, uniformly in time, remains close to the corresponding quasi-periodic solutions family of the cubic nonlinear Schrödinger, which are still defined in H^s. More precisely, we prove that their difference lies in H^{s-4} and it is of order O(c^-2). This behaviour is genuinely nonlinear: while such a uniform in time approximation already fails at the linear level, nonlinear interactions enable the dynamics of the two models to match globally in time. To the best of our knowledge, this provides the first global in time approximation results between KG and NLS in the compact setting. To establish the existence of small-amplitude solutions, we focus on the dependence, on the speed of light c, of the standard KAM construction, developed by Kuksin and Pöschel (and after by Berti-Biasco-Procesi). In particular, we construct a KAM algorithm for the Klein-Gordon equation which, at each step, remains closely related to a KAM algorithm for the nonlinear Schrödinger equation, thereby establishing a connection between the Hamiltonians of the two equations at every iteration. In particular, a new nontrivial difficulty arises: since the asymptotics of the frequencies depend on c, the classical tools for the analysis of the non-resonant sets must be revised, requiring the development of new methods. Chapters 2 and 3 focus on the defocusing Calogero-Moser derivative NLS equation (CMNLS), where D = -i ∂_x and Π denotes the Riesz-Szegö projection onto non-negative Fourier modes. The results obtained in these chapters draw on ideas and suggestions provided by P. Gérard. In particular, in Chapter 2 we present preliminary results about the Lax-pair structure of the PDE and the associated Birkhoff coordinates, which are the main properties that we need to introduce in Chapter 3 a Tao-type Gauge transformation G, following the classical strategy developed to prove a smoothing result for the Benjamin-Ono equation. In particular, we establish smoothing properties of the Birkhoff map, with a high-frequency approximation constructed from the transform G, and show that their difference is regularizing for some positive order τ(s). Finally, in the main theorem we prove a global well-posedness result for any initial data u_0 in H^s. Moreover, we show that, after applying the Gauge transform G, after the Tao-Gauge transformation, the solution is approximated by an explicit linear model derived from the Birkhoff coordinates of the equation, and the approximation error is globally bounded in time, in the higher regularity norm H^{s+τ(s)}. Methodologically, this thesis combines spectral analysis, Hamiltonian Normal Form and KAM techniques, Lax-pair integrability, and Birkhoff transformations to establish these two results on global in time approximation in compact domains, in regimes where dispersive decay is absent and recurrence phenomena dominate. In the following Introduction, we present an overview of the literature about the two problems and the techniques involved. In Section 1 we recall the most relevant known results about the non-relativistic limit of the Klein-Gordon equation. In Section 2 we present an overview about the KAM theory for PDEs. Finally, in Section 3, we introduce some known results about the CMNLS equation, recalling the derivation of the equation and well-posedness results; then we present the example of the smoothing properties of the Benjamin-Ono equation, which served as the basis for the method we developed to prove the smoothing result about the CMNLS equation.