Conservation laws, modulation and the emergence of universal forms.

Phase modulation has been a tool used for many years to describe system behaviour about periodic wavetrain solutions. The method has been shown to lead to well known partial differential equations (PDEs) in many contexts, describing how slowly varying perturbations to the phase of the solution behave. Typically these calculations can become involved and analyses are generally only valid for the one system consider and results cannot be transferred. This thesis discusses a format in which phase modulation leads to PDEs that emerge in a universal format, that is with coefficients determined by the wavetrain itself and the geometric properties of the system. The transition from simpler models emerging from the phase dynamics to more complex models is marked through the criticality of derivatives of the conservation laws associated with the system evaluated along the wavetrain considered. The analysis is also very general, so that the asymptotics need only be done once to apply to a large family of physical problems. We begin by discussing this method for single phase wavetrains, or more generally relative equilibrium, in the context of systems generated by a Lagrangian. By considering the modulation of single phased relative equilibrium for suitable scales, the Whitham equations are shown to emerge with a form explicitly related to the system's conservation laws. Furthermore, when the linearisation of the Whitham system develops zero characteristics (referred to as criticality), it can be shown that the modulation leads to the universal emergence of the Kadomtsev- Petviashvili (KP) and Boussinesq models. These models have coefficients which take the form of derivatives of the conservation law components evaluated on the relative equilibrium solution, as well as those that arise from a Jordan chain analysis. Additional degeneracies are shown to lead to fifth order KP and modified KP models of a similar universal form. The discussion is then shifted to the case of relative equilibria generated by multiple symmetries in 1 + 1 dimensions. It will be demonstrated how the results of the single phase case generalise naturally to additional phases, with criticality moving from scalar conditions to those of tensors. By undertaking similar analyses to the single phase, one obtains the multiphase Whitham equations, and depending on the number of zero characteristics its linearisation possesses the scalar Korteweg-de Vries (KdV), two-way Boussinesq and a new equation emerge instead. These analyses are supplemented with examples in stratified hydrodynamics and a coupled Nonlinear Schrodinger model. The final part of this thesis focusses on criticality conditions in multiphase modulations outside of the zero characteristic conditions. In particular, the aim is complete the derivation of all possible codimension 2 (equations requiring two conditions to be met simultaneously) and codimension 3 equations from the modulation approach. This recovers several well known equations, such as the modified KdV, fifth order KdV and higher order KdV equations, but also results in some novel equations such as the modified two-way Boussinesq and sixth order two-way Boussinesq equations. Examples of how these equations emerge are then given, revisiting the stratified shallow water and coupled Nonlinear Schrodinger equations to do so, demonstrating how the conditions for each reduction may be met and how the coefficients may be obtained