# Integrable boundary conditions for quad equations, open boundary reductions and integrable mappings

@article{Caudrelier2020IntegrableBC, title={Integrable boundary conditions for quad equations, open boundary reductions and integrable mappings}, author={Vincent Caudrelier and Peter H. van der Kamp and Cheng Zhang}, journal={arXiv: Mathematical Physics}, year={2020} }

In the context of integrable partial difference equations on quad-graphs, we introduce the notion of open boundary reductions as a new means to construct discrete integrable mappings and their invariants. This represents an alternative to the well-known periodic reductions. The construction deals with well-posed initial value problems for quad equations on quad-graphs restricted to a strip. It relies on the so-called double-row monodromy matrix and gives rise to integrable mappings. To obtain…

## 4 Citations

### Classification of Integrable Boundary Equations for Integrable Systems on Quad-Graphs

- MathematicsInternational Mathematics Research Notices
- 2022

In the context of integrable systems on quad-graphs, the boundary consistency around a half of a rhombic dodecahedron, as a companion notion to the three-dimensional consistency around a cube, was…

### Classification of integrable boundary equations for integrable quad-graph systems

- Mathematics
- 2021

In the context of integrable systems on quad-graphs, the boundary consistency around a half of a rhombic dodecahedron, as a companion notion to the three-dimensional consistency around a cube, was…

### A New Class of Integrable Maps of the Plane: Manin Transformations with Involution Curves

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2021

For cubic pencils we define the notion of an involution curve. This is a curve which intersects each curve of the pencil in exactly one non-base point of the pencil. Involution curves can be used to…

### Generalised Manin transformations and QRT maps

- MathematicsJournal of Computational Dynamics
- 2021

Manin transformations are maps of the plane that preserve a pencil of cubic curves. We generalise to maps that preserve quadratic, and certain quartic and higher degree pencils, and show they are…

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