Problems in Nonlinear Wave Propagation A Walk in Physics from Plasmas to Bose-Einstein Condensates with some Examples of Unifying Themes in Nature
Waves are a phenomenon that can be found virtually everywhere in nature. A first description of wave propagation can be given in the linear limit, but the nonlinear regime of propagation is of the utmost importance, also in view of possible applications in several scientific fields. In the course of this work, nonlinear wave propagation in physical systems from plasmas interacting with super-intense laser light to Bose-Einstein condensates (BEC) has been investigated, making use of the analogies brought to light by the mathematical modelization of such different systems. In the case of laser-plasma interactions, the main problem is the propagation of electromagnetic waves through a plasma. The nonlinear character is due to the high laser intensity which sets the plasma electrons into relativistic motion and exerts a force strongly perturbing their equilibrium density distribution. This deeply modifies the physics of the propagation leading to effects like self-induced transparency or the generation of plasma-field structures. Self-induced transparency is originated by the relativistic quiver motion of the plasma electrons and allows light to propagate through plasmas with a density so high that light propagation would classically be impossible. We have studied the problem of a threshold for induced transparency via an exact analytical investigation which has led furthermore to an exact description of the structures (electron depletion regions and light filaments) generated in the plasma as a consequence of the interaction, for both high and low density plasmas. The physics behind the generation of these structures can be described, in the weakly nonlinear limit, by the nonlinear Schrödinger equation (NLS), one of the fundamental nonlinear equations of physics. The importance and effectiveness of analytical investigations has been demonstrated by an analysis of the NLS equation generalized to the case of multi-dimensional non conservative systems (Ginzburg-Landau equation) and applied to the description of a scheme for the amplification of laser pulses (the chirped pulse amplification scheme). Furthermore, the same mathematical structure of the NLS equation describes the physics of BEC, systems of bosonic atoms that have undergone a phase transition such that they occupy the same ground state. It is the wave nature of matter which brings to light deep analogies with the nonlinear classical physics of optics and we have made of these analogies a tool for investigating certain aspects of the nature of a condensate. Once more, the mathematical modeling of physical phenomena has revealed new features of the underlying physics.
nonlinear Schroedinger equation