The study of Bose-Einstein condensates (BECs) has represented a core subject of physics for decades. Recently however, experiments have demonstrated a fundamentally distinct, inherently nonequilibriated class of BEC from photonic quasiparticles known as exciton-polaritons (polaritons). These photonic condensates are free to gain or lose energy as a part of their dynamics, and are thus not constrained to tend towards thermodynamic equilibrium. This greatly increases their pattern forming capabilities, but in turn severely complicates their theoretical treatment. The dynamical theory of these nonconservative condensates is an emerging field which resides at the intersection of the theories of nonequilibrium pattern formation, nonlinear wave dynamics and condensed matter theory. In this thesis I describe several contributions to this theory of $nonconservative$ quantum hydrodynamics, and towards the argument that it truly represents a paradigmatic departure from the now mature theory of equilibrium quantum hydrodynamics.

The polariton is formed in an optical cavity: cavity photons excite and superpose with excitons in a solid state sample to form bosonic light-matter quasiparticles. However, photons are trapped in the cavity for finite times, and are thus continually lost. A key characteristic of the polariton condensate is thus that to be created or sustained, they must be fed by an optical pump, which can take on any incident geometry, and which can be resonant or nonresonant with the natural frequency of the optical cavity. As a result, understanding the dynamical and structural implications of different forcing scenarios is fundamentally important for these systems, and exploring these scenarios is a major theme of this work.

In the first part of the thesis I focus on the role of pumping geometry on the dynamical behaviours and structural forms that can emerge. First, I show that an annular pumping geometry can lead to the spontaneous formation of stable multiply charged vortices, fundamental topological structures which have long been sought but are understood to be dynamically unstable even when imprinted and externally trapped in equilibrium BECs. The spontaneous formation is shown to come from the excitation of ring dark solitons in the early condensation, which are in this scenario dynamically unstable to breakup into vortices. I then show how the closed geometry of the forcing causes the stable binding of like-signed vortices via particle flux forces. It is shown that the topological charge limit on a multiply charged vortex formed this way is set by a Kelvin-Helmholtz instability, the first example of such an instability in a nonconservative condensate system. The acoustic properties of the multiply charged vortex are also considered, as they are found to emit topological charge dependent density waves. Links to analogue gravity and the process of quasinormal ringing are made.

I then elucidate the importance of the temporal symmetries imposed by the pump forcing. In particular, I show that the combination of non-resonant and resonant forcing generically leads to a fundamental breathing behaviour resulting from frustration between the incommensurate U$(1)$ phase symmetry of nonresonant forcing and the $Zn$ symmetry of $nth$ order resonant forcing. The most severe frustration is that between the U$(1)$ and $Z2$ symmetries, a case which I thus give special attention. In particular, I introduce a new solitary structure in this regime, a breathing ring dark soliton which represents a fundamental localized excitation of the extended condensate under this maximal phase frustration, forming spontaneously during the condensation process in a nonequilibrium analogue of the Kibble-Zurek mechanism. I also study the instability of vortices in this regime, which I show are unstable to self-slicing into dark solitons (Ising domain walls), the opposite transformation known to equilibria condensates, in which dark solitons are unstable to breakup into vortices via snake-instability. I then study the pattern forming abilities in a condensate with a radially dependent degree of phase-bistability, introducing a family of breather patterns which spontaneously break rotational symmetry in favor of polygonal spatial symmetries, the order of which can be tuned.

Finally, the inherent nonequilibration of the polariton condensate makes it a natural setting to consider the problem of turbulence. I introduce a process by which tuning the distances between a grid of pump spots allows for the formation of a nondecaying turbulent state of tunable average inter-vortex spacing. I show that this allows for the continuous tuning of quantum turbulence from the well known regime of superfluid turbulence (well separated vortices) into that of strong turbulence (separation of the order of a healing length), and into the theoretical regime of quantum weak turbulence, in which vortices have mean separations below the healing length and cores become destructured. I also discuss the possibility of observing the signatures of turbulence in polariton condensate experiments.