Electron dynamics in laser-driven one- and two-electron atoms and ions are probed through numerical integration of the full -dimensional, time-dependent Schro"dinger Equation. Physical observables, such as ionization rates and the spectrum of harmonics generated by the system, are extracted from the wavefunction. For an atom in a one-colour, linearly polarized laser field, wavepackets ejected at certain times will recoillide with the core, generating harmonics of the driving field. The harmonic spectrum is calculated using HELIUM and used to benchmark those from the time-dependent R-matrix (TDRM) method and a single-active-electron (SAE) code. HELIUM and TDRM results agree to within 20% while SAE calculations agree in the high-frequency region of the spectrum. The two-photon double ionization (TPDI) cross section of helium is calculated for frequencies 40 eV<ω< 54 a.u. In this frequency range, sequential ionization requires the absorption of three photons while non-sequential ionization is a two-photon process so that the latter mechanism is dominant for fields with low intensity. The cross section is shown to be sensitive to pulse duration, especially near the 54 eV threshold. Calculations using the HELIUM code provide TPDI cross sections accurate to a few percent across the whole frequency range. Extension of the method to treat one-electron atoms and ions in orthogonally polarized, two-colour laser fields are described. An orthogonally polarized, two-colour field will steer ionized wavepackets in the continuum. The likelihood of recollision, and the recolliding wavepackets velocity are significantly changed. These effects can be seen in the harmonic spectrum. Such a two-colour scheme has been proposed as a way to image complex molecular orbitals. The harmonic spectrum from model atomic systems in such fields is calculated. It is observed that the harmonic spectrum is sensitive to the symmetry of the electron orbitals. Thus, for the 2p state even harmonics are significantly stronger than those in the 2s case.

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This thesis concerns two distinct problems from the mathematical analysis of complex materials. One of the central themes is variational principles, which are widely used to understand physical phenomena, and one of the central aims of this type of work is to provide rigorous mathematical support to known results from physical literature. The first problem studied originates from micromagnetic theory (continuum theory for magnetism) and is that of domain wall motion in ferromagnetic nanowires. The magnetization dynamics is governed by a partial differential equation known as the Landau-Lifshitz-Gilbert equation. In this thesis, we present a new asymptotic method for understanding the dynamics of a domain wall in a thin (effectively one-dimensional) magnetic nanowire under the influence of an applied magnetic field or electronic spin current. We describe two flavours of solution: travelling waves, and oscillating solutions, as well as the transitions between them, by the use of perturbation expansions. We also provide a rigorous proof of the existence of travelling-wave solutions to this problem by the application of the implicit function theorem. We then present and analyze a heuristically derived model for the motion of vortex domain walls containing a micromagnetic singularity in thicker wires where the one-dimensional approximation fails. The second problem studied is that of phase transitions in biaxial nematic liquid crystals. Here, we study a variational theory based on the Onsager functional: a free energy functional derived from statistical mechanics of a liquid composed of anisotropic particles. We characterize the phase diagram of a biaxial liquid by considering three separate regimes: the weak-biaxial interaction regime (where the potential interaction between neighbouring molecules is close to uniaxial), the low-temperature (strong interaction) regime, and the near-isotropic regime. The global phase diagram is then inferred from these three analyses.

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This doctoral dissertation has considered several models to study the nonlinear dynamics of ion-acoustic electrostatic solitary and shock waves in a variety of non-Maxwellian plasmas using a fluid equation approach. The properties of ion acoustic solitary waves in both magnetised and unmagnetised three-dimensional collisionless plasmas containing excess superthermal electrons are investigated from first principles. With higher superthermality, the phase speed and dispersion of the waves are shown to be suppressed, and nonlinearity increases, leading to structures with smaller amplitudes and narrower widths. In magnetised plasmas the solitary wave width decreases with increased magnetic field strength. When dissipation is taken into account, using the fluid kinematic viscosity, electrostatic shock solutions can be found. Shocks become larger and narrower with stronger magnetic fields, higher superthermality and higher viscosity. The role of positrons is also explored. Both conservative and dissipative models are reviewed, in magnetised and unmagnetised plasma. Solitary wave frequency and phase speed decreases and structures become smaller and narrower when the positron concentration increases, and they increase in amplitude and width if the electron/positron temperature ratio is increased. Shock excitation amplitudes increase with increasing positron concentration, while widths decrease. A hybrid distribution function is employed combining a Cairns-type nonthermal form with the Tsallis theory for nonextensive thermodynamics. Arbitrary amplitude ion acoustic solitary wave dynamics in a two-component plasma are investigated, and the valid range of the function examined. Solitary waves are shown to exist within a narrow range of allowable Mach numbers. Both positive and negative potential structures are found, and coexistence may occur. Finally an unmagnetised collisionless electron-ion plasma model is proposed, featuring a non-Maxwellian-trapped electron distribution, modelled by a kappa distribution function combined with a Schamel distribution. With enhanced superthermality, the amplitude and width of solitary waves decreases. Shock waves are shown to be possible and their dynamics discussed.

The control and detection of quantum correlations in three distinct mesoscopic systems is discussed. The entanglement dynamics between the momentum modes of an intracavity BEC and the motional modes of a mechanical end-mirror is studied. Although the interaction between these mesoscopic systems is indirect and mediated by a cavity field, mirror-atom entanglement is found to arise early on in the dynamics. Optimal control techniques are applied to modulate the driving laser such that mirror-atom entanglement is generated over longer timescales. In the following chapters, a versatile framework of characterising multipartite entanglement with entanglement witnesses in rotationally invariant states is derived. Together with the twirling map, this framework can be applied to arbitrary spin systems and is demonstrated for the XXZ spin-1/2 model, where long-range multipartite entanglement is shown to be present close to a critical point and indicates breaking of the symmetries associated with the phase transition of the model. The final chapters detail how the quantum polarisation spectroscopy measurement scheme, applied to a system of trapped bosons in an optical lattice, can be used to characterise the phase transitions of the Bose-Hubbard model and the extended Bose-Hubbard model by analysing atomic collective correlations. This all-optical non-demolition measurement scheme is additionally applied to detect and characterise the ground states of atoms trapped by a superlattice potential, formed by counter-propagating beams.