The Ablowitz-Ladik system on the natural numbers with certain linearizable boundary conditions

Abstract

We solve the initial-boundary value problem (IBVP) for the Ablowitz–Ladik system on the natural numbers with certain linearizable boundary conditions. We do so by employing a nonlinear method of images, namely, by extending the scattering potential to all integers in such a way that the extended potential satisfies certain symmetry relations. Using these extensions and the solution of the initial value problem (IVP), we then characterize the symmetries of the discrete spectrum of the scattering problem, and we show that discrete eigenvalues in the IBVP appear in octets, as opposed to quartets in the IVP. Furthermore, we derive explicit relations between the norming constants associated with symmetric eigenvalues, and we identify a new kind of linearizable IBVP. Finally, we characterize the soliton solutions of these IBVPs, which describe the soliton reflection at the boundary of the lattice.

Keywords:

• solitons
• integrable systems
• boundary value problems
• nonlinear method of images

AMS Subject Classifications:

• 37K15
• 35Q51
• 35R10
• 35G30

Acknowledgements

We thank I. Habibullin for interesting discussions, B. Prinari for valuable suggestions about expressing the normalization constant C _−∞ in terms of the scattering data, and one of the anonymous reviewers for helpful comments on solutions associated with self-symmetric eigenvalues. This work was partially supported by NSF under grant number DMS-0908399.