Soliton solutions of noncommutative integrable systems

• Main Author: Sooman, Craig
• Published: University of Glasgow 2010
• Subjects: 510; QA Mathematics
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.508006

This thesis is concerned with solutions of noncommutative integrable systems where the noncommutativity arises through the dependent variables in either the hierarchy or Lax pair generating the equation. Both Chapters 1 and 2 are entirely made up of background material and contain no new material. Furthermore, these chapters are concerned with commutative equations. Chapter 1 outlines some of the basic concepts of integrable systems including historical attempts at finding solutions of the KdV equation, the Lax method and Hirota's direct method for finding multi-soliton solutions of an integrable system. Chapter 2 extends the ideas in Chapter 1 from equations of one spatial dimension to equations of two spatial dimensions, namely the KP and mKP equations. Chapter 2 also covers the concepts of hierarchies and Darboux transformations. The Darboux transformations are iterated to give multi-soliton solutions of the KP and mKP equations. Furthermore, this chapter shows that multi-soliton solutions can be expressed as two types of determinant: the Wronskian and the Grammian. These determinantal solutions are then verified directly. In Chapter 3, the ideas detailed in the preceding chapters are extended to the noncommutative setting. We begin by outlining some known material on quasideterminants, a noncommutative KP hierarchy containing a noncommutative KP equation, and also two families of solutions. The two families of solutions are obtained from Darboux transformations and can be expressed as quasideterminants. One family of solutions is termed ``quasiwronskian'' and the other ``quasigrammian'' as both reduce to Wronskian and Grammian determinants when their entries commute. Both families of solutions are then verified directly. The remainder of Chapter 3 is original material, based on joint work with Claire Gilson and Jon Nimmo. Building on some known results, the solutions obtained from the Darboux transformations are specified as matrices. These solutions have interesting interaction properties not found in the commutative setting. We therefore show various plots of the solutions illustrating these properties. In Chapter 4, we repeat all of the work of Chapter 3 for a noncommutative mKP equation. The material in this chapter is again based on joint work with Claire Gilson and Jon Nimmo and is mainly original. The original material in Chapters 3 and 4 appears in \cite{gilson:nimmo:sooman2008} and in \cite{gilson:nimmo:sooman2009}. Chapter 5 builds on the work of Chapters 3 and 4 and is concerned with exponentially localised structures called dromions, which are obtained by taking the determinant of the matrix solutions of the noncommutative KP and mKP equations. For both equations, we look at a three-dromion structure from which we then perform a detailed asymptotic analysis. This aymptotic forms show interesting interaction properties which are demonstrated by various plots. This chapter is entirely the author's own work. Chapter 6 presents a summary and conclusions of the thesis.