# The Ablowitz-Ladik system on a finite set of integers

## TL;DR

This paper develops a method to solve initial-boundary value problems for the integrable Ablowitz-Ladik system on a finite set of integers using a discrete analogue of the Fokas method, involving a matrix Riemann-Hilbert problem.

## Contribution

It extends the unified transform method to the Ablowitz-Ladik system on finite sets, providing a way to handle boundary conditions via spectral functions and the global relation.

## Key findings

- Solution expressed as a matrix Riemann-Hilbert problem with explicit $(n,t)$ dependence.
- Characterization of boundary values through the global relation.
- Discussion of linearizable boundary conditions.

## Abstract

We show how to solve initial-boundary value problems for integrable nonlinear differential-difference equations on a finite set of integers. The method we employ is the discrete analogue of the unified transform (Fokas method). The implementation of this method to the Ablowitz-Ladik system yields the solution in terms of the unique solution of a matrix Riemann-Hilbert problem, which has a jump matrix with explicit $(n,t)$-dependence involving certain functions referred to as spectral functions. Some of these functions are defined in terms of the initial value, while the remaining spectral functions are defined in terms of two sets of boundary values. These spectral functions are not independent but satisfy an algebraic relation called global relation. We analyze the global relation to characterize the unknown boundary values in terms of the given initial and boundary values. We also discuss the linearizable boundary conditions.