The thesis is centred on nonlinear Partial Differential Equations that appear in the modelling of shallow water waves. In particular, the Riemann problem for such models is investigated and the results, to the best of my knowledge, are original. In a case where the result appear in a previous work, acknowledgements and references are made accordingly. The thesis has not been submitted for other degree or diploma in any other university. The content of this thesis is in two parts. Part I consists of general background information about water wave theory and summary of the main research results. Part II consists of the following papers written during the work of the thesis:

The Brio system is a two-by-two system of conservation laws arising as a simplified model in ideal magnetohydrodynamics (MHD). The system has the formIt was found in previous works that the standard theory of hyperbolic conservation laws does not apply to this system since the characteristic fields are not genuinely nonlinear on the set v = 0. As a consequence, certain Riemann problems have no weak solutions in the traditional class of functions of bounded variation. It was argued in [8] that in order to solve the system, singular solutions containing Dirac masses along the shock waves might have to be used. Solutions of this type were exhibited in [11,23], but uniqueness was not obtained.In the current work, we introduce a nonlinear change of variables which makes it possible to solve the Riemann problem in the framework of the standard theory of conservation laws. In addition, we develop a criterion which leads to an admissibility condition for singular solutions of the original system, and it can be shown that admissible solutions are unique in the framework developed here.

The thesis is centred on nonlinear Partial Differential Equations that appear in the modelling of shallow water waves. In particular, the Riemann problem for such models is investigated and the results, to the best of my knowledge, are original. In a case where the result appear in a previous work, acknowledgements and references are made accordingly. The thesis has not been submitted for other degree or diploma in any other university. The content of this thesis is in two parts. Part I consists of general background information about water wave theory and summary of the main research results. Part II consists of the following papers written during the work of the thesis:

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