# New integrable boundary conditions for the Ablowitz-Ladik model: from Hamiltonian formalism to nonlinear mirror image method

## TL;DR

This paper develops new integrable boundary conditions for the Ablowitz-Ladik model using Hamiltonian formalism, connecting it with nonlinear methods to construct explicit solutions, including multisolitons, on finite and half-infinite lattices.

## Contribution

It introduces a novel extension of the mirror image method for time-dependent boundary conditions, linking Hamiltonian, zero curvature, and B"acklund transformation approaches.

## Key findings

- Derived new boundary conditions generalizing Robin conditions.
- Extended mirror image method to time-dependent boundaries.
- Constructed explicit multisoliton solutions with examples.

## Abstract

Using Sklyanin's classical theory of integrable boundary conditions, we use the Hamiltonian approach to derive new integrable boundary conditions for the Ablowitz-Ladik model on the finite and half infinite lattice. In the case of half infinite lattice, the special and new emphasis of this paper is to connect directly the Hamiltonian approach, based on the classical $r$-matrix, with the zero curvature representation and B\"acklund transformation approach that allows one to implement a nonlinear mirror image method and construct explicit solutions. It is shown that for our boundary conditions, which generalise (discrete) Robin boundary conditions, a nontrivial extension of the known mirror image method to what we call {\it time-dependent boundary conditions} is needed. A careful discussion of this extension is given and is facilitated by introducing the notion of intrinsic and extrinsic picture for describing boundary conditions. This gives the specific link between Sklyanin's reflection matrices and B\"acklund transformations combined with folding, {\it in the case of non-diagonal reflection matrices}. All our results reproduce the known Robin boundary conditions setup as a special case: the diagonal case. Explicit formulas for constructing multisoliton solutions on the half-lattice with our time-dependent boundary conditions are given and some examples are plotted.