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Using Sklyanin's classical theory of integrable boundary conditions, we usethe Hamiltonian approach to derive new integrable boundary conditions for theAblowitz-Ladik model on the finite and half infinite lattice. In the case ofhalf infinite lattice, the special and new emphasis of this paper is to connectdirectly the Hamiltonian approach, based on the classical $r$-matrix, with thezero curvature representation and B\"acklund transformation approach thatallows one to implement a nonlinear mirror image method and construct explicitsolutions. It is shown that for our boundary conditions, which generalise(discrete) Robin boundary conditions, a nontrivial extension of the knownmirror image method to what we call {\it time-dependent boundary conditions} isneeded. A careful discussion of this extension is given and is facilitated byintroducing the notion of intrinsic and extrinsic picture for describingboundary conditions. This gives the specific link between Sklyanin's reflectionmatrices and B\"acklund transformations combined with folding, {\it in the caseof non-diagonal reflection matrices}. All our results reproduce the known Robinboundary conditions setup as a special case: the diagonal case. Explicitformulas for constructing multisoliton solutions on the half-lattice with ourtime-dependent boundary conditions are given and some examples are plotted.

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