The analysis of PDEs arising from the Korteweg-de Vries hierarchies

Maphanga, Rivoningo
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In this dissertation, we study the hierarchy commonly defined as an infinite sequence of partial differential equations which begins with the Korteweg-deVries equation and its modified version. We look at how Lie point symmetries and conservation laws of each of these hierarchies aid the solving of higher order partial differential equations through associations and transformations. An important feature of these hierarchies is their highly nonlinear property. In this regard, obtaining solutions for the members of these hierarchies poses a great problem, where in the past, it was impossible to calculate solutions. In this study, we establish a method to allow for the construction of new solutions to the full hierarchy. We begin by determining point symmetries and conserved vectors of each hierarchy, then proceed to testing association between the obtained symmetries and conserved vectors, and finally using transformations to construct solutions
A dissertation submitted in partial fulfilment of the requirements for the degree Masters of Science (by coursework and research report), for the Faculty of Science, University of the Witwatersrand, 2021