Integrable hierarchies in the Lax/zero-curvature formalism

Abstract

This thesis focusses on the development of (1+1)-dimensional integrable hierarchies in both the classical and quantum settings via the Lax/zero-curvature picture, where the underlying Poisson structure is found through the use of a classical or quantum R-matrix. After setting the scene by using the non-linear Schrodinger and isotropic Landau-Lifshitz models as examples of the standard approach to constructing hierarchies in this picture, the focus shifts to two more recent developments: equal-space Poisson structures (and the resulting spatially conserved quantities and Lax pairs); and quantum Lax pairs, where previously only the quantum Lax matrix (the spatial component) was considered. The non-linear Schrodinger and isotropic Landau-Lifshitz models (or analogous quantum spin chains) are then used as examples for these recent developments to compare against the familiar results.