# Initial-boundary value problems associated with the Ablowitz-Ladik system

## TL;DR

This paper demonstrates how to analyze initial-boundary value problems for the Ablowitz-Ladik system, an integrable nonlinear differential-difference equation, using the unified transform method, and expresses solutions via matrix Riemann-Hilbert problems.

## Contribution

It applies the Fokas method to the Ablowitz-Ladik system, deriving solution representations and boundary value elimination techniques for discrete integrable equations.

## Key findings

- Solutions expressed via matrix Riemann-Hilbert problems
- Global relations for boundary data derived
- Method applicable to discrete nonlinear Schrödinger and mKdV equations

## Abstract

We employ the Ablowitz-Ladik system as an illustrative example in order to demonstrate how to analyze initial-boundary value problems for integrable nonlinear differential-difference equations via the unified transform (Fokas method). In particular, we express the solutions of the integrable discrete nonlinear Schr\"{o}dinger and integrable discrete modified Korteweg-de Vries equations in terms of the solutions of appropriate matrix Riemann-Hilbert problems. We also discuss in detail, for both the above discrete integrable equations, the associated global relations and the process of eliminating of the unknown boundary values.