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The Ablowitz-Ladik system on the natural numbers with certain linearizable boundary conditions. (English) Zbl 1220.37061

The authors solve the initial-boundary value problem for the Ablowitz-Ladik system on the natural numbers with certain linearizable boundary conditions 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, they characterize the symmetries of the discrete spectrum of the scattering problem and show that discrete eigenvalues in the linearizable boundary conditions appear in octets, as opposed to quartets in the linearizable boundary conditions. Furthermore, the authors derive explicit relations between the norming constants associated with symmetric eigenvalues. Finally, they characterize the soliton solutions of these linearizable boundary conditions which describe the soliton reflection at the boundary of the lattice.

MSC:

37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
35Q51 Soliton equations
35R10 Partial functional-differential equations
35G30 Boundary value problems for nonlinear higher-order PDEs
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References:

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