There is a well established connection between one parameter Lie groups of transformations and conservation laws for differential equations. In this thesis, we construct conservation laws via the invariance and multiplier approach based on the wellknown result that the Euler-Lagrange operator annihilates total divergences. This technique will be applied to some plasma physics models. We show that the recently developed notion of the association between Lie point symmetry generators and conservation laws lead to double reductions of the underlying equation and ultimately to exact/invariant solutions for higher-order nonlinear partial di erential equations viz., some classes of Schr odinger and KdV equations.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2012. / Unable to load abstract.

PhD., Faculty of Science, University of the Witwatersrand, 2011 / The construction of conserved vectors using Noether’s theorem via a knowledge of a Lagrangian (or via the recently developed concept of partial Lagrangians) is well known. The formulae to determine these for higher-order flows is somewhat cumbersome and becomes more so as the order increases. We carry out these for a class of fourth, fifth and sixth order PDEs. In the latter case, we involve the fifth-order KdV equation using the concept of ‘weak’ Lagrangians better known for the third-order KdV case. We then consider the case of a mixed ‘high-order’ equations working on the Shallow Water Wave and Regularized Long Wave equations. These mixed type equations have not been dealt with thus far using this technique. The construction of conserved vectors using Noether’s theorem via a knowledge of a Lagrangian is well known. In some of the examples, our focus is that the resultant conserved flows display some previously unknown interesting ‘divergence properties’ owing to the presence of the mixed derivatives. We then analyse the conserved flows of some multi-variable equations that arise in Relativity. In addition to a larger class of conservation laws than those given by the isometries or Killing vectors, we may conclude what the isometries are and that these form a Lie subalgebra of the Noether symmetry algebra. We perform our analysis on versions of the Vaidya metric yielding some previously unknown information regarding the corresponding manifold. Lastly, with particular reference to this metric, we also show the variations that occur for the unknown functions. We discuss symmetries of classes of wave equations that arise as a consequence of the Vaidya metric. The objective of this study is to show how the respective geometry is responsible for giving rise to a nonlinear inhomogeneous wave equation as an alternative to assuming the existence of nonlinearities in the wave equation due to physical considerations. We find Lie and Noether point symmetries of the corresponding wave equations and give some reductions. Some interesting physical 4 conclusions relating to conservation laws such as energy, linear and angular momenta are also determined. We also present some interesting comparisons with the standard wave equations (on a ‘flat geometry’). Finally, we pursue the nature of the flow of a third grade fluid with regard to its underlying conservation laws. In particular, the fluid occupying the space over a wall is considered. At the surface of the wall, suction or blowing velocity is applied. By introducing a velocity field, the governing equations are reduced to a class of PDEs. A complete class of conservation laws for the resulting equations are constructed and analysed using the invariance properties of the corresponding multipliers/characteristics.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, in requirement for the degree Doctor of Philosophy, Johannesburg, 2013. / The (1+3) dimensional wave and Klein-Gordon equations are constructed using the covariant d'Alembertian operator on several spacetimes of interest. Equations on curved geometry inherit the nonlinearities of the geometry. These equations display interesting properties in a number of ways. In particular, the number of symmetries and therefore, the conservation laws reduce depending on how curved the manifold is. We study the symmetry properties and conservation laws of wave equations on Freidmann-Robertson-Walker, Milne, Bianchi, and de Sitter universes. Symmetry structures are used to reduce the number of unknown functions, and hence contribute to nding exact solutions of the equations. As expected, properties of reduction procedures using symmetries, variational structures and conservation laws are more involved than on the well known at (Minkowski) manifold.

Ke, Ting. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 30-32). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction and Main results --- p.1 / Chapter 2 --- Preliminaries --- p.7 / Chapter 3 --- Finite Speed of Propagation Property --- p.11 / Chapter 4 --- Proof of the Main Results --- p.19 / Chapter 4.1 --- Proof of Theorem 1.0.1 --- p.19 / Chapter 4.2 --- Proof of Theorem 1.0.2 --- p.24 / Chapter 5 --- Discussions --- p.26 / Bibliography --- p.30

<p>“In many interesting papers on discrete velocity models (DVMs), authors postulate from the beginning that the finite velocity space with "good" properties is given and only after this step they study the Discrete Boltzmann Equation. Contrary to this approach, our aim is not to study the equation, but to discuss all possible choices of finite phase spaces (sets) satisfying this type of "good" restrictions. Due to the velocity discretization it is well known that it is possible to have DVMs with "spurious" summational invariants (conservation laws which are not linear combinations of physical invariants). Our purpose is to give a method for constructing normal models (without spurious invariants) and to classify all normal plane models with small number of velocities (which usually appear in applications). On the first step we describe DKMs as algebraic systems. We introduce for this an abstract discrete model (ADM) which is defined by a matrix of reactions (the same as for the concrete model). This matrix contains as rows all vectors of reactions describing the "jump" from a pre-reaction state to a new reaction state. The conservation laws corresponding to the many-particle system are uniquely determined by the ADM and do not depend on the concrete realization. We find the restrictions on ADM and then we give a general method of constructing concrete normal models (using the results on ADMs). Having the general algorithm, we consider in more detail, the particular cases of models with mass and momentum conservation (inelastic lattice gases with pair collisions) and models with mass, momentum and energy conservation (elastic lattice gases with pair collisions).”</p>

“In many interesting papers on discrete velocity models (DVMs), authors postulate from the beginning that the finite velocity space with "good" properties is given and only after this step they study the Discrete Boltzmann Equation. Contrary to this approach, our aim is not to study the equation, but to discuss all possible choices of finite phase spaces (sets) satisfying this type of "good" restrictions. Due to the velocity discretization it is well known that it is possible to have DVMs with "spurious" summational invariants (conservation laws which are not linear combinations of physical invariants). Our purpose is to give a method for constructing normal models (without spurious invariants) and to classify all normal plane models with small number of velocities (which usually appear in applications). On the first step we describe DKMs as algebraic systems. We introduce for this an abstract discrete model (ADM) which is defined by a matrix of reactions (the same as for the concrete model). This matrix contains as rows all vectors of reactions describing the "jump" from a pre-reaction state to a new reaction state. The conservation laws corresponding to the many-particle system are uniquely determined by the ADM and do not depend on the concrete realization. We find the restrictions on ADM and then we give a general method of constructing concrete normal models (using the results on ADMs). Having the general algorithm, we consider in more detail, the particular cases of models with mass and momentum conservation (inelastic lattice gases with pair collisions) and models with mass, momentum and energy conservation (elastic lattice gases with pair collisions).”

Classical kinetic theory of gases is based on the Boltzmann equation (BE) which describes the evolution of a system of particles undergoing collisions preserving mass, momentum and energy. Discretization methods have been developed on the idea of replacing the original BE by a finite set of nonlinear hyperbolic PDEs corresponding to the densities linked to a suitable finite set of velocities. One open problem related to the discrete BE is the construction of normal (fulfilling only physical conservation laws) discrete velocity models (DVMs). In many papers on DVMs, authors postulate from the beginning that a finite velocity space with such "good" properties is given, and after this step, they study the discrete BE. Our aim is not to study the equations for DVMs, but to discuss all possible choices of finite phase spaces (sets) satisfying this type of "good" restrictions. We start by introducing the most general class of discrete kinetic models (DKMs) and then, develop a general method for the construction and classification of normal DKMs. We apply this method in the particular cases of DVMs of the inelastic BE (where we show that all normal models can be explicitly described) and elastic BE (where we give a complete classification of normal models up to 9 velocities). Using our general approach to DKMs and our results on normal DVMs for a single gas, we develop a method for the construction of the most natural (from physical point of view) subclass of normal DVMs for binary gas mixtures. We call such models supernormal models (SNMs). We apply this method and obtain SNMs with up to 20 velocities and their spectrum of mass ratio. Finally, we develop a new method that can lead, by symmetric transformations, from a given normal DVM to extended normal DVMs. Many new normal models can be constructed in this way, and we give some examples to illustrate this.

This thesis examines the importance of energy and momentum consistency in subgrid-scale parameterization for climate models. It is divided into two parts according to the two aspects of the problem that are investigated, namely the importance of momentum conservation alone and the consistency between energy and momentum conservation. The first part addresses the importance of momentum conservation alone. Using a zonally-symmetric model, it is shown that violating momentum conservation in the parameterization of gravity wave drag leads to large errors and non-robustness of the response to an imposed radiative perturbation in the middle atmosphere. Using the Canadian Middle Atmosphere Model, a three-dimensional climate model, it is shown that violating momentum conservation, by allowing gravity wave momentum flux to escape through the model lid, leads to large errors in the mean climate when the model lid is placed at 10 hPa. When the model lid is placed at 0.001 hPa the errors due to nonconservation are minimal. When the 10 hPa climate is perturbed by idealized ozone depletion in the southern hemisphere, nonconservation is found to significantly alter the polar temperature and surface responses. Overall, momentum conservation ensures a better agreement between the 10 hPa and the 0.001 hPa climates. The second part addresses the self-consistency of energy and momentum conservation. Using Hamiltonian geophysical fluid dynamics, pseudoenergy and pseudomomentum wave-activity conservation laws are derived for the subgrid-scale dynamics. Noether’s theorem is used to derive a relationship between the wave-activity fluxes, which represents a generalization of the first Eliassen-Palm theorem. Using multiple scale asymptotics a theoretical framework for subgrid-scale parameterization is built which consistently conserves both energy and momentum and respects the second law of thermodynamics. The framework couples a hydrostatic resolved-scale flow to a non-hydrostatic subgrid-scale flow. The transfers of energy and momentum between the two scales are understood using the subgrid-scale wave-activity conservation laws, whose relationships with the resolved-scale dynamics represent generalized non-acceleration theorems. The derived relationship between the wave-activity fluxes — which represents a generalization of the second Eliassen-Palm theorem — is key to ensuring consistency between energy and momentum conservation. The framework includes a consistent formulation of heating and entropy production due to kinetic energy dissipation.

This thesis examines the importance of energy and momentum consistency in subgrid-scale parameterization for climate models. It is divided into two parts according to the two aspects of the problem that are investigated, namely the importance of momentum conservation alone and the consistency between energy and momentum conservation. The first part addresses the importance of momentum conservation alone. Using a zonally-symmetric model, it is shown that violating momentum conservation in the parameterization of gravity wave drag leads to large errors and non-robustness of the response to an imposed radiative perturbation in the middle atmosphere. Using the Canadian Middle Atmosphere Model, a three-dimensional climate model, it is shown that violating momentum conservation, by allowing gravity wave momentum flux to escape through the model lid, leads to large errors in the mean climate when the model lid is placed at 10 hPa. When the model lid is placed at 0.001 hPa the errors due to nonconservation are minimal. When the 10 hPa climate is perturbed by idealized ozone depletion in the southern hemisphere, nonconservation is found to significantly alter the polar temperature and surface responses. Overall, momentum conservation ensures a better agreement between the 10 hPa and the 0.001 hPa climates. The second part addresses the self-consistency of energy and momentum conservation. Using Hamiltonian geophysical fluid dynamics, pseudoenergy and pseudomomentum wave-activity conservation laws are derived for the subgrid-scale dynamics. Noether’s theorem is used to derive a relationship between the wave-activity fluxes, which represents a generalization of the first Eliassen-Palm theorem. Using multiple scale asymptotics a theoretical framework for subgrid-scale parameterization is built which consistently conserves both energy and momentum and respects the second law of thermodynamics. The framework couples a hydrostatic resolved-scale flow to a non-hydrostatic subgrid-scale flow. The transfers of energy and momentum between the two scales are understood using the subgrid-scale wave-activity conservation laws, whose relationships with the resolved-scale dynamics represent generalized non-acceleration theorems. The derived relationship between the wave-activity fluxes — which represents a generalization of the second Eliassen-Palm theorem — is key to ensuring consistency between energy and momentum conservation. The framework includes a consistent formulation of heating and entropy production due to kinetic energy dissipation.