Initial value problems for quad equations

TL;DR

This paper presents a method for constructing well-posed initial value problems for quad equations on certain types of quad-graphs, focusing on rhombic embeddable graphs, with implications for both integrable and non-integrable cases.

Contribution

It introduces a new method to establish well-posed initial value problems specifically for rhombic embeddable quad-graphs, expanding the understanding of initial value problem construction.

Findings

Method is effective for rhombic embeddable quad-graphs.

Not all quad-graphs admit well-posed initial value problems.

The approach applies to both integrable and non-integrable equations.

Abstract

We describe a method to construct well-posed initial value problems for not necessarily integrable equations on not necessarily simply connected quad-graphs. Although the method does not always provide a well-posed initial value problem (not all quad-graphs admit well-posed initial value problems) it is effective in the class of rhombic embeddable quad-graphs.