Some Integrable Finite-Dimensional Systems And Their Continuous Counterparts

The discrete spectral problem of Ablowitz--Ladik is considered in the case in which the potential has a finite support of length L. The spectral transform is explicitly computed and a recurrence relation on the length L for computing it in L algebraic step is given. This spectral transform can be used to generate via the scattering method a finite dimensional version of the dynamical systems associated to the Ablowitz--Ladik spectral problem. A special case in which the potential can be constraint to evolve in time on a semi-line is proposed. The truncated soliton, i.e. the potential obtained by putting to zero the one soliton outside an interval of length L is examined in detail. The sufficient and necessary condition for having a soliton contained in the truncated soliton solution is derived. Finally, the continuous counterpart of these finite--dimensional systems is considered. The spectral transform is derived via a Riccati equation and the special case of the truncated soliton is..