The Korteweg-de Vries equation and its homologues : a comparative analysis
The Korteweg-de Vries equation (KdV) is a partial differential equation which has some remarkable mathematical properties. Furthermore, it also appears as a useful model in a great many physical situations. Thus, although it was originally obtained as an approximation in fluid dynamics, it was reinterpreted as a canonical field theory for weakly dispersive and weakly nonlinear systems. This reinterpretation led to the hypothesis that the properties of the KdV could be understood in terms of a balance between the competing effects of dispersion and nonlinearity. Alternatives to the KdV were proposed on the basis that their dispersive properties were physically and mathematically preferable to those of the KdV. The use of dispersion, which is a linear concept, as a criterion for predicting the properties of these nonlinear equations was examined in an earlier thesis by Abbas. By introducing a general class of equations which includes the KdV and all its proposed alternatives as special cases, Abbas investigated in detail the predictions based on the dispersion relation and compared them with the actual properties of the equation, particularly in regard to the existence of solitary waves. He found little correlation and some contradictions and concluded that the idea of a balance between nonlinearity and dispersion is not useful way of understanding these equations. It is clear, therefore, that we must develop other criteria to obtain this understanding. In this thesis we continue this investigation by looking at other properties of the class of equations introduced by Abbas which are relevant to the KdV. The general question which we are considering is whether the properties of the KdV are unique in this class and if so how can we decide this a priori, i.e., from the equation and its elementary solutions. A prerequisite for tackling this problem is to establish whether the embedding of the KdV in this class is reasonable, i.e., that these equations can indeed be considered as homologues of the KdV. Thus, it is necessary to establish well-posedness, the existence of solitary wave and other elementary solutions and the existence of other properties such as, for example, conservation laws. These are the specific questions that we consider in this thesis. To make the thesis self-contained we begin with a comprehensive review of the KdV and its main alternative, the regularized long wave equation, together with the work of Abbas. This comprises the first part of the thesis and puts our own contribution in its proper perspective. The second part of the thesis contains our own contribution and begins with a completion of the analysis of solitary waves begun by Abbas. We next partition the general class into five equivalence classes and establish well-posedness for three of them and existence for a fourth. Finally, we show that all equations have at least two conservation laws, some of the equations have at most three conservation laws. These results enable us to conclude that this class of equations is a reasonable one in which to investigate the question referred to above. The thesis ends with a résumé and suggests avenues for continuing this investigation.