Nonlinear physics of ultracold quantum gases and optical fields in media

Ultracold quantum gases and optical fields in nonlinear media are highly controllable systems for exploring nonlinear wave physics and quantum phenomena. Their tunable geometry and interaction strengths make it possible to investigate both static and dynamic properties under a wide range of conditions. Solitary waves, for example, have been predicted and observed in both settings. The underlying interaction physics is strongly shaped by dimensionality, which also governs the role of beyond-mean-field effects. As a result, dimensional crossover plays a central role in determining both the behaviour of these systems. This thesis addresses three interconnected themes within this broad class of physical systems. First, we develop an approximation in quantum scattering theory, called the on-shell approximation of the two-body T -matrix, establishing explicit relations between low-energy scattering observables and effective couplings in one, two, and three dimensions, and we benchmark the method against solvable models. Second, we investigate the dynamics and stability of matter-wave and optical solitons in quasi-one-dimensional geometries, combining variational dimensional reductions techniques with numerical simulations. These studies clarify how confinement influences soliton stability, collapse thresholds, and transmission across potential barriers, and they connect closely with an experimental realization of ultracold Cesium in an optical lattice. Third, we analyze fluctuation effects within a beyond-mean-field theory approach in Josephson junctions and superconducting electrodynamics. We show how quantum fluctuations modify dynamical features of the junction, like the Josephson oscillation frequency, and critical self-trapping threshold, and how to describe the interaction with electromagnetic fields determining the fields penetration depths in superconductors. Together, these results show how scattering theory, soliton dynamics, and fluctuation corrections can be consistently treated within effective low-dimensional frameworks, and how they are present across multiple physical systems.