Quasinormal modes of non-relativistic systems : analogue gravity with the optical soliton

The quasinormal modes of a black hole are characteristic features of the underlying black hole spacetime. Developing our understanding of black hole quasinormal modes promises insights into both classical and quantum gravity. This thesis develops and extends the theory of quasinormal modes in a number of non-relativistic settings, advancing their role in characterizing systems described by wave equations. Beginning with the classical framework of damped oscillators and black hole quasinormal modes, we identify challenges in spectral computation and stability, as well as prospects for black hole spectroscopy through gravitational wave observations. Motivated by these challenges, we introduce a new analogue gravity system based on optical solitons in nonlinear fibres. Within a generalized nonlinear Schrödinger framework, we establish that solitons support quasinormal modes, derive their spectra, and solve the initial value problem to connect quasinormal modes with ringdown phenomena. By constructing an analogy with black holes—exact in extremal Schwarzschild–de Sitter spacetimes and extendable via the Mashhoon approximation—we show that solitons can serve as simulators of black hole ringdown physics, enabling experimental tests of black hole spectral instability. We complement this with the first numerical method tailored to non-relativistic quasinormal modes, adapting the compactified hyperboloidal approach to compute Schrödinger spectra and uncover preliminary evidence of spectral instability. Finally, we generalize the concept of quasinormal modes so that it may be applied to higher-order wave equations, facilitating the formulation of quasinormal modes in Lorentz-violating theories. Together, these results lay the foundations of a broad research programme, spanning analogue experiments, numerical methods, and theoretical extensions, aimed at deepening our understanding of quasinormal modes beyond their traditional relativistic domain.