Zhang, Cheng (2013). Continuous and quadgraph integrable models with a boundary: Reflection maps and 3Dboundary consistency. (Unpublished Doctoral thesis, City University London)

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Abstract
This thesis is focusing on boundary problems for various classical integrable schemes.
First, we consider the vector nonlinear Schrodinger (NLS) equation on the halfline. Using a Backlund transformation method which explores the folding symmetry of the system, classes of integrable boundary conditions (BCs) are derived. These BCs coincide with the linearizable BCs obtained using the unified transform method developed by Fokas. The notion of integrability is argued by constructing an explicit generating function for conserved quantities. Then, by adapting a mirror image technique, an inverse scattering method with an integrable boundary is constructed in order to obtain Nsoliton solutions on the halfline, i.e. Nsoliton reflections. An interesting phenomenon of transmission between different components of vector solitons before and after interacting with the boundary is demonstrated.
Next, in light of the fact that the solitonsoliton interactions give rise to YangBaxter maps, we realize that the solitonboundary interactions that are extracted from the Nsoliton reflections can be translated into maps satisfying the settheoretical counterpart of the quantum reflection equation. Solutions of the settheoretical reflection equation are referred to as reflection maps. Both the YangBaxter maps and the reflection maps guarantee the factorization of the solitonsoliton and solitonboundary interactions for vector NLS solitons on the halfline.
Indeed, reflection maps represent a novel mathematical structure. Basic notions such as parametric reflection maps, their graphic representations and transfer maps are also introduced. As a natural extension, this object is studied in the context of quadrirational YangBaxter maps, and a classification of quadrirational reflection maps is obtained.
Finally, boundaries are added to discrete integrable systems on quadgraphs. Triangle configurations are used to discretize quadgraphs with boundaries. Relations involving vertices of the triangles give rise to boundary equations that are used to described BCs. We introduce the notion of integrable BCs by giving a threedimensional boundary consistency as a criterion for integrability. By exploring the correspondence between the quadrirational YangBaxter maps and the socalled ABS classification, we also show that quadrirational reflection maps can be used as a systematic tool to generate integrable boundary equations for the equations from the ABS classification.
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