Soliton solutions of ultradiscrete integrable systems

In recent years, ultradiscrete integrable systems, in which both independent and dependent variables are discretized, have attracted much attention. In this thesis, we show how to obtain all line soliton solutions of (2+1)-dimensional ultradiscrete soliton systems from determinant solutions of discrete soliton systems by taking an ultradiscrete limit. Taking an ultradiscrete limit of determinant solutions with non-negativity, we obtain Casorati determinant-like solutions. Starting from Grammian (Gram-type determinant), we obtain another expression of τ-functions, which leads to a perturbed form after the expansion of Grammian. These two different forms are essentially equivalent, i.e., it is possible to transform a form into the other form by a simple combinatorics. In a Casorati determinant-like expression, there is a big advantage of constructing all possible line soliton solutions easily. Using ultradiscrete determinant-like solutions, we study the details of line soliton interactions of the ultradiscrete two-dimensional Toda lattice (2DTL) equation

Linear superposition and interaction of Wronskian solutions to an extended (2+1)-dimensional KdV equation

The main purpose of this work is to discuss an extended KdV equation, which can provide some physically significant integrable evolution equations to model the propagation of two-dimensional nonlinear solitary waves in various science fields. Based on the bilinear Bäcklund transformation, a Lax system is constructed, which guarantees the integrability of the introduced equation. The linear superposition principle is applied to homogeneous linear differential equation systems, which plays a key role in presenting linear superposition solutions composed of exponential functions. Moreover, some special linear superposition solutions are also derived by extending the involved parameters to the complex field. Finally, a set of sufficient conditions on Wronskian solutions is given associated with the bilinear Bäcklund transformation. The Wronskian identities of the bilinear KP hierarchy provide a direct and concise way for proving the Wronskian determinant solution. The resulting Wronskian structure generates $N$-soliton solutions and a few of special Wronskian interaction solutions, which enrich the solution structure of the introduced equation

Soliton solutions of noncommutative integrable systems

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

Soliton solutions of noncommutative integrable systems

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.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Direct linearisation of discrete and continuous integrable systems: The KP hierarchy and its reductions

The thesis is concerned with the direct linearisation of discrete and continuous integrable systems, which aims to establish a unified framework to study integrable discrete and continuous nonlinear equations, and to reveal the underlying structure behind them. The idea of the direct linearisation approach is to connect a nonlinear equation with a linear integral equation. By introducing an infinite-dimensional matrix structure to the linear integral equation, we are able to study various nonlinear equations in the same class and their interlinks simultaneously, as well as the associated integrability properties. Meanwhile, the linear integral equation also provides a general class of solutions to those nonlinear equations, in which the well-known soliton-type solutions to those nonlinear equations can be recovered very easily. In the thesis, we consider discrete and continuous integrable equations associated with scalar linear integral equations. The framework is illustrated by three-dimensional models including the discrete and continuous Kadomtsev--Petviashvili-type equations as well as the discrete-time two-dimensional Toda lattice, and their dimensional reductions which result in a huge class of two-dimensional discrete and continuous integrable systems

Polinomios biortogonales y sus generalizaciones: una perspectiva desde los sistemas integrables

La conexión existente entre los polinomios ortogonales y otras ramas de la matemática, la física o la ingeniería es verdaderamente asombrosa. Además, no hay mejor prueba de la utilidad de estos que el propio crecimiento, avance perpetuo y generalización en diversas direcciones de lo que se entendía por polinomio ortogonal en los albores de la teoría. Conforme el concepto se fue generalizando, también fueron evolucionando las técnicas para su estudio, algunas de estas claramente influenciadas por aquellas disciplinas matemáticas con las que iban surgiendo conexiones. La perspectiva que esta tesis adopta frente a los polinomios ortogonales es un ejemplo de este tipo de influencias, compartiendo herramientas y entrelazandose con la teoría de los sistemas integrables. Una posición privilegiada en esta tesis la ocuparían las matrices de Gram semi in nitas; cada cual asociada a una forma sesquilineal adaptada al tipo de biortogonalidad en cuestión. A estas matrices se les impondrán una serie de condiciones cuyo objeto sería el de garantizar la existencia y unicidad de las secuencias biortogonales asociadas a las mismas. El siguiente paso consistiría en buscar simetrías de estas matrices de Gram. Existen dos razones por las que este esfuerzo resulta ventajoso. En primer lugar, cada simetría encontrada podría traducirse en propiedades de las secuencias biortogonales, por ejemplo: una estructura Hankel de la matriz es equivalente a gozar de la recurrencia a tres términos de los polinomios ortogonales; la simetría propia de las matrices asociadas a pesos clásicos (Hermite, Laguerre, Jacobi) implica la existencia del operador diferencial lineal de segundo orden de que los polinomios clásicos son solución; etc..