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The Korteweg-DeVries equation and its homologues III analytical structure

Mehmet, M. (1990). The Korteweg-DeVries equation and its homologues III analytical structure. (Unpublished Doctoral thesis, City, University of London)

Abstract

This thesis is the third in a series of studies on the Korteweg-de Vries equation (KdV) and its homologues, the objective being to understand its distinguished position when embedded in a class of similar equations.

Now the KdV is a partial differential equation which is well-known to have 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 first study, which was undertaken by Abbas, was to test the hypothesis described above that dispersion is a qseful criterion for constructing nonlinear equations. By introducing a general class of equations which Includes the KdV and all its proposed alternatives as special cases, he 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 a useful way of understanding these equations. This meant that other criteria must be developed to obtain this understanding.

The criteria we are looking for would have to account for the existence of families of solitary waves in the general class and, in the case of the KdV, for solitons. However, before doing this it was Important to establish the mathematical validity of the equations, l.e. well-posedness and existence of conservation laws. This was carried out by El-Sherblny in the second study of the series. By partitioning the set of equations into equivalence classes, he proved existence for most of the equations and well-posedness for some. He also showed that, with the exception of the KdV, all the equations have at least two and at most three conservation laws.

At the time that this third study was started interest was focussed on the integrabllity of nonlinear evolution equations and, through the Palnlevé conjectures, this was reformulated in terms of the analytic structure of the solutions of these equations. It seemed natural, therefore, to look at the Abbas' class of equations from this point of view as the major objective. In addition, we critically examine the structure of the solitary waves themselves in order to answer the question of when a solitary wave is a soliton.

The first part of this thesis contains the introduction and relevant reviews of the inverse scattering method, integrability and the work of Abbas and El-Sherblny.

The second part of the thesis contains our main contributions and we begin by obtaining all the similarity reductions to ordinary differential equations for the general class of partial differential equations using one-parameter Lie groups. We derive the singularity structure of the general solutions of the similarity equations and use this analysis to Initiate a classification of third order nonlinear ordinary differential equations. Next we obtain the singularity structure of classes of general solutions of' the partial differential equations directly in terms of Laurent-type expansions. These results are compared with those obtained via symmetry groups and equations which are or are not Palnlevé-type identified. We also look for special cases of the general solution which may be restricted solitons. We do not find any, and, in the case of the regularised long wave equation, we prove that it does not have any. Finally, we develop a classification of the solitary waves of the general class and use this to develop necessary criteria for a solitary wave to be a soliton.

The thesis ends with a resumé and suggests avenues for continuing this investigation.

Publication Type: Thesis (Doctoral)
Subjects: Q Science > QA Mathematics
Departments: School of Science & Technology > Mathematics
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