Zhang, Cheng (2013).
*Continuous and quad-graph integrable models with a boundary: Reflection maps and 3D-boundary consistency*.
(Unpublished Doctoral thesis, City University London)

## 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 N-soliton solutions on the half-line, i.e. N-soliton 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 soliton-soliton interactions give rise to Yang-Baxter maps, we realize that the soliton-boundary interactions that are extracted from the N-soliton reflections can be translated into maps satisfying the set-theoretical counterpart of the quantum reflection equation. Solutions of the set-theoretical reflection equation are referred to as reflection maps. Both the Yang-Baxter maps and the reflection maps guarantee the factorization of the soliton-soliton and soliton-boundary interactions for vector NLS solitons on the half-line.

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 Yang-Baxter maps, and a classification of quadrirational reflection maps is obtained.

Finally, boundaries are added to discrete integrable systems on quad-graphs. Triangle configurations are used to discretize quad-graphs 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 three-dimensional boundary consistency as a criterion for integrability. By exploring the correspondence between the quadrirational Yang-Baxter maps and the so-called 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.

Publication Type: | Thesis (Doctoral) |
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Subjects: | Q Science > QA Mathematics |

Departments: | Doctoral Theses School of Mathematics, Computer Science & Engineering > Mathematics Doctoral Theses > School of Mathematics, Computer Science and Engineering Doctoral Theses |