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.