In this work we investigate granular chains, which are one-dimensional systems of discrete macroscopic particles interacting via the intrinsically nonlinear Hertz law. Such systems support the propagation of solitary waves (SWs), which are non-dispersive, mobile bundles of energy. A comprehensive analysis into the dynamical behaviour of these systems and the properties of SW propagation is presented, and several interesting new results are obtained. First, we find that the transition to the quasi-equilibrium (QEQ) phase in granular chains can be manipulated by altering the material properties of the system. We further use these results to develop a novel shock absorption device with a predictable and tunable frequency response, making it useful also for energy harvesting applications. Second, we show for the first time that granular chains with various nonlinearities of the contact potential can achieve thermal equilibrium at sufficiently long times, and thus QEQ is an intermediate phase of these systems. We characterize the equilibrium phase by deriving approximate distribution functions for grain velocity and kinetic energy and system kinetic energy in a microcanonical ensemble of interacting particles. As a by-product, we derive the equilibrium specific heat, and a size-dependent correction term, for such systems. We also show how these ideas extend to heterogeneous systems such as diatomic, tapered, and random-mass chains. Furthermore, we look closely at the transition to equilibrium by using statistical tests to show that the long-term dynamics is ergodic, and by examining the behaviour of various correlation functions close to the onset of the transition. Third, we solve a highly nonlinear, fourth-order wave equation that models the continuum theory of long-wavelength pulses in weakly compressed, homogeneous granular chains with a general power-law contact interaction, to characterize all travelling wave solutions admitted by the equation. This involves deriving conservation laws admitted by the wave equation, followed by a modified energy analysis. We find that the wave equation admits various types of travelling wave solutions, including SW solutions as well as nonlinear periodic wave solutions. Not only have the SW solutions not appeared before in the literature on granular chains, but they are also a new addition to the literature on SWs in general.

All kinematic conservation laws along with their symmetry transformation properties are derived for the system of magnetohydrodynamic equations governing incompressible viscous plasmas (or any other conducting fluid) in which the dynamic and magnetic viscosities are constant. Reductions of this system under translation symmetries, axial rotation symmetries, and helical symmetries are considered. For each reduced system, all kinematic conservation laws and point symmetries are obtained. The results yields several new conservation laws which are expected to be relevant for physical applications of magnetohydrodynamics.

The effects of magnetic dilution and applied pressure on frustrated spinels GeNi2O4, GeCo2O4, and NiAl2O4 are reported. Dilution was achieved by substitution of Mg2+ in place of magnetically active Co2+ and Ni2+ ions. Large values of the percolation thresholds were found in GeNi(2-x)MgxO4. Specifically, pc1 = 0.74 and pc2 = 0.65 in the sub-networks associated with the triangular and kagome planes, respectively. This anomalous behaviour may be explained by the kagome and triangular planes behaving as coupled networks, also know as a network of networks. In simulations of coupled lattices that form a network of networks, similar anomalous percolation threshold values have been found. In addition, at dilution levels above x=0.30, there is a T^2 dependency in the magnetic heat capacity which may indicate two dimensional spin glass behaviour. Applied pressures in the range of 0 GPa to 1.2 GPa yield a slight decrease in ordering temperature for both the kagome and triangular planes. In GeCo(2-x)MgxO4, the long range magnetic order is more robust with a percolation threshold of pc=0.448. Similar to diluted nickel germanate, at low temperatures, a T^2 magnetic heat capacity contribution is present which indicates a shift from a 3D ordered state to a 2D spin glass state in the presence of increased dilution. Dynamic magnetic susceptibility data indicate a change from canonical spin glass to a cluster glass behaviour. In addition, there is a non-linear increase in ordering temperature with applied pressure in the range P = 0 to 1.0 GPa. A spin glass ground state was observed in Ni(1-x)MgxAl2O4 for (x=0 to 0.375). Analysis of dynamic magnetic susceptibility data yield a characteristic time of tau* = 1.0x10^(-13) s, which is indicative of canonical spin glass behaviour. This is further corroborated by the linear behaviour of the magnetic specific heat contribution. However, the increasing frequency dependence of the freezing temperature suggests a trend towards spin cluster glass formation.

Copper arsenite CuAs2O4 and Copper antimonite CuSb2O4 are S=1/2 (Cu2+ 3d9 electronic configuration) quasi-one-dimensional quantum spin-chain compounds. Both compounds crystallize with tetragonal structures containing edge sharing CuO6 octahedra chains which experience Jahn-Teller distortions. The basal planes of the octahedra link together to form CuO2 ribbon-chains which harbor Cu2+ spin-chains. These compounds are magnetically frustrated with competing nearest-neighbour and next-nearest-neighbour intrachain spin-exchange interactions. Despite the similarities between CuAs2O4 and CuSb2O4, they exhibit very different magnetic properties. In this thesis work, the physical properties of CuAs2O4 and CuSb2O4 are investigated using a variety of experimental techniques which include x-ray diffraction, magnetic susceptibility measurements, heat capacity measurements, Raman spectroscopy, electron paramagnetic resonance, neutron diffraction, and dielectric capacitance measurements. CuAs2O4 exhibits dominant ferromagnetic nearest-neighbour and weaker antiferromagnetic next-nearest-neighbour intrachain spin-exchange interactions. The ratio of the intrachain interactions amounts to Jnn/Jnnn = -4.1. CuAs2O4 was found to order with a ferromagnetic groundstate below TC = 7.4 K. An extensive physical characterization of the magnetic and structural properties of CuAs2O4 was carried out. Under the effect of hydrostatic pressure, CuAs2O4 was found to undergo a structural phase transition at 9 GPa to a new spin-chain structure. The structural phase transition is accompanied by a severe alteration of the magnetic properties. The high-pressure phase exhibits dominant ferromagnetic next-nearest-neighbour spin-exchange interactions and weaker ferromagnetic nearest-neighbour interactions. The ratio of the intrachain interactions in the high-pressure phase was found to be Jnn/Jnnn = 0.3. Structural and magnetic characterizations under hydrostatic pressure are reported and a relationship between the structural and magnetic properties was established. CuSb2O4 orders antiferromagnetically below TN = 1.8 K with an incommensurate helicoidal magnetic structure. CuSb2O4 is characterized by ferromagnetic nearest-neighbour and antiferromagnetic next-nearest-neighbour spin-exchange interactions with Jnn/Jnnn = -1.8. A (H, T) magnetic phase diagram was constructed using low-temperature magnetization and heat capacity measurements. The resulting phase diagram contains multiple phases as a consequence of the strong intrachain magnetic frustration. Indications of ferroelectricity were observed in the incommensurate antiferromagnetic phase.